↑

Properties

Label 12-378e6-1.1-c3e6-0-2
Degree 1212
Conductor 2.917×10152.917\times 10^{15}
Sign 11
Analytic cond. 1.23068×1081.23068\times 10^{8}
Root an. cond. 4.722574.72257
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 12·4-s − 5·5-s + 8·7-s + 16·8-s + 30·10-s + 29·11-s + 20·13-s − 48·14-s − 144·16-s + 38·17-s + 57·19-s − 60·20-s − 174·22-s + 14·23-s + 267·25-s − 120·26-s + 96·28-s − 362·29-s − 88·31-s + 288·32-s − 228·34-s − 40·35-s + 384·37-s − 342·38-s − 80·40-s + 432·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 0.447·5-s + 0.431·7-s + 0.707·8-s + 0.948·10-s + 0.794·11-s + 0.426·13-s − 0.916·14-s − 9/4·16-s + 0.542·17-s + 0.688·19-s − 0.670·20-s − 1.68·22-s + 0.126·23-s + 2.13·25-s − 0.905·26-s + 0.647·28-s − 2.31·29-s − 0.509·31-s + 1.59·32-s − 1.15·34-s − 0.193·35-s + 1.70·37-s − 1.45·38-s − 0.316·40-s + 1.64·41-s + ⋯

Functional equation

Λ(s)=((26⋅318⋅76)s/2 ΓC(s)6 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((26⋅318⋅76)s/2 ΓC(s+3/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 26⋅318⋅762^{6} \cdot 3^{18} \cdot 7^{6}
Sign: 11
Analytic conductor: 1.23068×1081.23068\times 10^{8}
Root analytic conductor: 4.722574.72257
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 26⋅318⋅76, ( :[3/2]6), 1)(12,\ 2^{6} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 2.9475084362.947508436
L(12)L(\frac12) ≈\approx 2.9475084362.947508436
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 (1+pT+p2T2)3 ( 1 + p T + p^{2} T^{2} )^{3}
3 1 1
7 1−8T−22pT2+139p2T3−22p4T4−8p6T5+p9T6 1 - 8 T - 22 p T^{2} + 139 p^{2} T^{3} - 22 p^{4} T^{4} - 8 p^{6} T^{5} + p^{9} T^{6}
good5 1+pT−242T2−1291T3+31349T4+94946T5−3671531T6+94946p3T7+31349p6T8−1291p9T9−242p12T10+p16T11+p18T12 1 + p T - 242 T^{2} - 1291 T^{3} + 31349 T^{4} + 94946 T^{5} - 3671531 T^{6} + 94946 p^{3} T^{7} + 31349 p^{6} T^{8} - 1291 p^{9} T^{9} - 242 p^{12} T^{10} + p^{16} T^{11} + p^{18} T^{12}
11 1−29T−2564T2+60061T3+4759283T4−58883294T5−5900798285T6−58883294p3T7+4759283p6T8+60061p9T9−2564p12T10−29p15T11+p18T12 1 - 29 T - 2564 T^{2} + 60061 T^{3} + 4759283 T^{4} - 58883294 T^{5} - 5900798285 T^{6} - 58883294 p^{3} T^{7} + 4759283 p^{6} T^{8} + 60061 p^{9} T^{9} - 2564 p^{12} T^{10} - 29 p^{15} T^{11} + p^{18} T^{12}
13 (1−10T+6508T2−43337T3+6508p3T4−10p6T5+p9T6)2 ( 1 - 10 T + 6508 T^{2} - 43337 T^{3} + 6508 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} )^{2}
17 1−38T−10148T2+214318T3+64981502T4−384215870T5−350665294625T6−384215870p3T7+64981502p6T8+214318p9T9−10148p12T10−38p15T11+p18T12 1 - 38 T - 10148 T^{2} + 214318 T^{3} + 64981502 T^{4} - 384215870 T^{5} - 350665294625 T^{6} - 384215870 p^{3} T^{7} + 64981502 p^{6} T^{8} + 214318 p^{9} T^{9} - 10148 p^{12} T^{10} - 38 p^{15} T^{11} + p^{18} T^{12}
19 1−3pT−14838T2+254831T3+166435821T4−166961520T5−1338012994557T6−166961520p3T7+166435821p6T8+254831p9T9−14838p12T10−3p16T11+p18T12 1 - 3 p T - 14838 T^{2} + 254831 T^{3} + 166435821 T^{4} - 166961520 T^{5} - 1338012994557 T^{6} - 166961520 p^{3} T^{7} + 166435821 p^{6} T^{8} + 254831 p^{9} T^{9} - 14838 p^{12} T^{10} - 3 p^{16} T^{11} + p^{18} T^{12}
23 1−14T−24188T2+1111042T3+287732402T4−11473339118T5−3133578440789T6−11473339118p3T7+287732402p6T8+1111042p9T9−24188p12T10−14p15T11+p18T12 1 - 14 T - 24188 T^{2} + 1111042 T^{3} + 287732402 T^{4} - 11473339118 T^{5} - 3133578440789 T^{6} - 11473339118 p^{3} T^{7} + 287732402 p^{6} T^{8} + 1111042 p^{9} T^{9} - 24188 p^{12} T^{10} - 14 p^{15} T^{11} + p^{18} T^{12}
29 (1+181T+44385T2+9691225T3+44385p3T4+181p6T5+p9T6)2 ( 1 + 181 T + 44385 T^{2} + 9691225 T^{3} + 44385 p^{3} T^{4} + 181 p^{6} T^{5} + p^{9} T^{6} )^{2}
31 1+88T−19176T2+2427606T3−74706896T4−63836425952T5+24418143111607T6−63836425952p3T7−74706896p6T8+2427606p9T9−19176p12T10+88p15T11+p18T12 1 + 88 T - 19176 T^{2} + 2427606 T^{3} - 74706896 T^{4} - 63836425952 T^{5} + 24418143111607 T^{6} - 63836425952 p^{3} T^{7} - 74706896 p^{6} T^{8} + 2427606 p^{9} T^{9} - 19176 p^{12} T^{10} + 88 p^{15} T^{11} + p^{18} T^{12}
37 1−384T+6558T2+23609678T3−2205845910T4−979004345454T5+385902155257863T6−979004345454p3T7−2205845910p6T8+23609678p9T9+6558p12T10−384p15T11+p18T12 1 - 384 T + 6558 T^{2} + 23609678 T^{3} - 2205845910 T^{4} - 979004345454 T^{5} + 385902155257863 T^{6} - 979004345454 p^{3} T^{7} - 2205845910 p^{6} T^{8} + 23609678 p^{9} T^{9} + 6558 p^{12} T^{10} - 384 p^{15} T^{11} + p^{18} T^{12}
41 (1−216T+4638pT2−29937897T3+4638p4T4−216p6T5+p9T6)2 ( 1 - 216 T + 4638 p T^{2} - 29937897 T^{3} + 4638 p^{4} T^{4} - 216 p^{6} T^{5} + p^{9} T^{6} )^{2}
43 (1+363T+118869T2+51110485T3+118869p3T4+363p6T5+p9T6)2 ( 1 + 363 T + 118869 T^{2} + 51110485 T^{3} + 118869 p^{3} T^{4} + 363 p^{6} T^{5} + p^{9} T^{6} )^{2}
47 1−183T−263418T2+21765813T3+50364860037T4−1987643355060T5−5827318049035865T6−1987643355060p3T7+50364860037p6T8+21765813p9T9−263418p12T10−183p15T11+p18T12 1 - 183 T - 263418 T^{2} + 21765813 T^{3} + 50364860037 T^{4} - 1987643355060 T^{5} - 5827318049035865 T^{6} - 1987643355060 p^{3} T^{7} + 50364860037 p^{6} T^{8} + 21765813 p^{9} T^{9} - 263418 p^{12} T^{10} - 183 p^{15} T^{11} + p^{18} T^{12}
53 1−396T−171222T2+33593058T3+26399387130T4+3085597337094T5−6213949690744577T6+3085597337094p3T7+26399387130p6T8+33593058p9T9−171222p12T10−396p15T11+p18T12 1 - 396 T - 171222 T^{2} + 33593058 T^{3} + 26399387130 T^{4} + 3085597337094 T^{5} - 6213949690744577 T^{6} + 3085597337094 p^{3} T^{7} + 26399387130 p^{6} T^{8} + 33593058 p^{9} T^{9} - 171222 p^{12} T^{10} - 396 p^{15} T^{11} + p^{18} T^{12}
59 1−427T−20846T2+184632833T3−989648387pT4−9945164910982T5+19570132793727955T6−9945164910982p3T7−989648387p7T8+184632833p9T9−20846p12T10−427p15T11+p18T12 1 - 427 T - 20846 T^{2} + 184632833 T^{3} - 989648387 p T^{4} - 9945164910982 T^{5} + 19570132793727955 T^{6} - 9945164910982 p^{3} T^{7} - 989648387 p^{7} T^{8} + 184632833 p^{9} T^{9} - 20846 p^{12} T^{10} - 427 p^{15} T^{11} + p^{18} T^{12}
61 1+7pT−325797T2−168225036T3+60617986357T4+21366910723417T5−7516285932983282T6+21366910723417p3T7+60617986357p6T8−168225036p9T9−325797p12T10+7p16T11+p18T12 1 + 7 p T - 325797 T^{2} - 168225036 T^{3} + 60617986357 T^{4} + 21366910723417 T^{5} - 7516285932983282 T^{6} + 21366910723417 p^{3} T^{7} + 60617986357 p^{6} T^{8} - 168225036 p^{9} T^{9} - 325797 p^{12} T^{10} + 7 p^{16} T^{11} + p^{18} T^{12}
67 1+32T−507154T2+145892682T3+107477835646T4−39824972320442T5−19149017206317005T6−39824972320442p3T7+107477835646p6T8+145892682p9T9−507154p12T10+32p15T11+p18T12 1 + 32 T - 507154 T^{2} + 145892682 T^{3} + 107477835646 T^{4} - 39824972320442 T^{5} - 19149017206317005 T^{6} - 39824972320442 p^{3} T^{7} + 107477835646 p^{6} T^{8} + 145892682 p^{9} T^{9} - 507154 p^{12} T^{10} + 32 p^{15} T^{11} + p^{18} T^{12}
71 (1+395T+772713T2+166861721T3+772713p3T4+395p6T5+p9T6)2 ( 1 + 395 T + 772713 T^{2} + 166861721 T^{3} + 772713 p^{3} T^{4} + 395 p^{6} T^{5} + p^{9} T^{6} )^{2}
73 1−373T−1002964T2+125934813T3+757146856945T4−40185482983214T5−334578208592440487T6−40185482983214p3T7+757146856945p6T8+125934813p9T9−1002964p12T10−373p15T11+p18T12 1 - 373 T - 1002964 T^{2} + 125934813 T^{3} + 757146856945 T^{4} - 40185482983214 T^{5} - 334578208592440487 T^{6} - 40185482983214 p^{3} T^{7} + 757146856945 p^{6} T^{8} + 125934813 p^{9} T^{9} - 1002964 p^{12} T^{10} - 373 p^{15} T^{11} + p^{18} T^{12}
79 1−1364T+318072T2+576829470T3−209637930452T4−324379754397308T5+410997275277132367T6−324379754397308p3T7−209637930452p6T8+576829470p9T9+318072p12T10−1364p15T11+p18T12 1 - 1364 T + 318072 T^{2} + 576829470 T^{3} - 209637930452 T^{4} - 324379754397308 T^{5} + 410997275277132367 T^{6} - 324379754397308 p^{3} T^{7} - 209637930452 p^{6} T^{8} + 576829470 p^{9} T^{9} + 318072 p^{12} T^{10} - 1364 p^{15} T^{11} + p^{18} T^{12}
83 (1+77T+1084713T2+260668709T3+1084713p3T4+77p6T5+p9T6)2 ( 1 + 77 T + 1084713 T^{2} + 260668709 T^{3} + 1084713 p^{3} T^{4} + 77 p^{6} T^{5} + p^{9} T^{6} )^{2}
89 1−1233T−175281T2+162125424T3+290804803809T4+425032882464177T5−812862837191430722T6+425032882464177p3T7+290804803809p6T8+162125424p9T9−175281p12T10−1233p15T11+p18T12 1 - 1233 T - 175281 T^{2} + 162125424 T^{3} + 290804803809 T^{4} + 425032882464177 T^{5} - 812862837191430722 T^{6} + 425032882464177 p^{3} T^{7} + 290804803809 p^{6} T^{8} + 162125424 p^{9} T^{9} - 175281 p^{12} T^{10} - 1233 p^{15} T^{11} + p^{18} T^{12}
97 (1−590T+2642891T2−1009069940T3+2642891p3T4−590p6T5+p9T6)2 ( 1 - 590 T + 2642891 T^{2} - 1009069940 T^{3} + 2642891 p^{3} T^{4} - 590 p^{6} T^{5} + p^{9} T^{6} )^{2}
show more
show less
   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.86148471111733568546121503502, −5.52745225849444771746830944257, −5.22991046504957596186149501329, −5.17104654916587106817644210623, −5.09249209372309443201355917510, −4.68145162377822280577453766086, −4.62475228324082912094816400040, −4.23619399408778225734081085620, −4.23296080620778120181288038116, −3.97847843779030189653846359056, −3.87748138054753054719000194904, −3.34608442781561910538730058059, −3.32469621477230505613751270311, −3.28408527425718895105221551539, −2.90211616142841532217637152364, −2.54205317263844294492702781335, −2.32691300124844405972830248087, −1.95452282616591445564969828142, −1.83250347677178253703460338138, −1.49061274589125013420155109191, −1.32209824682547876094861521346, −0.968269276729676681173141192169, −0.60853727991463714428019050016, −0.57738221664454986667404265357, −0.42045621464521149117025341185, 0.42045621464521149117025341185, 0.57738221664454986667404265357, 0.60853727991463714428019050016, 0.968269276729676681173141192169, 1.32209824682547876094861521346, 1.49061274589125013420155109191, 1.83250347677178253703460338138, 1.95452282616591445564969828142, 2.32691300124844405972830248087, 2.54205317263844294492702781335, 2.90211616142841532217637152364, 3.28408527425718895105221551539, 3.32469621477230505613751270311, 3.34608442781561910538730058059, 3.87748138054753054719000194904, 3.97847843779030189653846359056, 4.23296080620778120181288038116, 4.23619399408778225734081085620, 4.62475228324082912094816400040, 4.68145162377822280577453766086, 5.09249209372309443201355917510, 5.17104654916587106817644210623, 5.22991046504957596186149501329, 5.52745225849444771746830944257, 5.86148471111733568546121503502

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.