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Properties

Label 12-378e6-1.1-c1e6-0-2
Degree 1212
Conductor 2.917×10152.917\times 10^{15}
Sign 11
Analytic cond. 756.159756.159
Root an. cond. 1.737331.73733
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 21·4-s + 5·5-s + 4·7-s − 56·8-s − 30·10-s + 11-s − 2·13-s − 24·14-s + 126·16-s + 4·17-s − 3·19-s + 105·20-s − 6·22-s + 7·23-s + 19·25-s + 12·26-s + 84·28-s + 5·29-s + 28·31-s − 252·32-s − 24·34-s + 20·35-s − 9·37-s + 18·38-s − 280·40-s + 12·41-s + ⋯
L(s)  = 1  − 4.24·2-s + 21/2·4-s + 2.23·5-s + 1.51·7-s − 19.7·8-s − 9.48·10-s + 0.301·11-s − 0.554·13-s − 6.41·14-s + 63/2·16-s + 0.970·17-s − 0.688·19-s + 23.4·20-s − 1.27·22-s + 1.45·23-s + 19/5·25-s + 2.35·26-s + 15.8·28-s + 0.928·29-s + 5.02·31-s − 44.5·32-s − 4.11·34-s + 3.38·35-s − 1.47·37-s + 2.91·38-s − 44.2·40-s + 1.87·41-s + ⋯

Functional equation

Λ(s)=((26⋅318⋅76)s/2 ΓC(s)6 L(s)=( Λ(2−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(2-s)\end{aligned}
Λ(s)=((26⋅318⋅76)s/2 ΓC(s+1/2)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+1/2)^{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: 756.159756.159
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 26⋅318⋅76, ( :[1/2]6), 1)(12,\ 2^{6} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.1268254191.126825419
L(12)L(\frac12) ≈\approx 1.1268254191.126825419
L(32)L(\frac{3}{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+T)6 ( 1 + T )^{6}
3 1 1
7 1−4T+2pT2−55T3+2p2T4−4p2T5+p3T6 1 - 4 T + 2 p T^{2} - 55 T^{3} + 2 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
good5 1−pT+6T2−T3+31T4−68T5+29T6−68pT7+31p2T8−p3T9+6p4T10−p6T11+p6T12 1 - p T + 6 T^{2} - T^{3} + 31 T^{4} - 68 T^{5} + 29 T^{6} - 68 p T^{7} + 31 p^{2} T^{8} - p^{3} T^{9} + 6 p^{4} T^{10} - p^{6} T^{11} + p^{6} T^{12}
11 1−T−6T2+103T3−83T4−32pT5+457pT6−32p2T7−83p2T8+103p3T9−6p4T10−p5T11+p6T12 1 - T - 6 T^{2} + 103 T^{3} - 83 T^{4} - 32 p T^{5} + 457 p T^{6} - 32 p^{2} T^{7} - 83 p^{2} T^{8} + 103 p^{3} T^{9} - 6 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}
13 1+2T−32T2−2pT3+730T4+230T5−10729T6+230pT7+730p2T8−2p4T9−32p4T10+2p5T11+p6T12 1 + 2 T - 32 T^{2} - 2 p T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 230 p T^{7} + 730 p^{2} T^{8} - 2 p^{4} T^{9} - 32 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
17 1−4T+9T2−92T3+58T4+20T5+5393T6+20pT7+58p2T8−92p3T9+9p4T10−4p5T11+p6T12 1 - 4 T + 9 T^{2} - 92 T^{3} + 58 T^{4} + 20 T^{5} + 5393 T^{6} + 20 p T^{7} + 58 p^{2} T^{8} - 92 p^{3} T^{9} + 9 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
19 1+3T−42T2−61T3+69pT4+726T5−27501T6+726pT7+69p3T8−61p3T9−42p4T10+3p5T11+p6T12 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
23 1−7T−24T2+127T3+1417T4−3484T5−22393T6−3484pT7+1417p2T8+127p3T9−24p4T10−7p5T11+p6T12 1 - 7 T - 24 T^{2} + 127 T^{3} + 1417 T^{4} - 3484 T^{5} - 22393 T^{6} - 3484 p T^{7} + 1417 p^{2} T^{8} + 127 p^{3} T^{9} - 24 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12}
29 1−5T−30T2+371T3−185T4−6020T5+44357T6−6020pT7−185p2T8+371p3T9−30p4T10−5p5T11+p6T12 1 - 5 T - 30 T^{2} + 371 T^{3} - 185 T^{4} - 6020 T^{5} + 44357 T^{6} - 6020 p T^{7} - 185 p^{2} T^{8} + 371 p^{3} T^{9} - 30 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12}
31 (1−14T+138T2−841T3+138pT4−14p2T5+p3T6)2 ( 1 - 14 T + 138 T^{2} - 841 T^{3} + 138 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 1+9T−21T2−268T3+1293T4+4875T5−42882T6+4875pT7+1293p2T8−268p3T9−21p4T10+9p5T11+p6T12 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 4875 p T^{7} + 1293 p^{2} T^{8} - 268 p^{3} T^{9} - 21 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}
41 1−12T−18T2+78T3+7470T4−24546T5−158105T6−24546pT7+7470p2T8+78p3T9−18p4T10−12p5T11+p6T12 1 - 12 T - 18 T^{2} + 78 T^{3} + 7470 T^{4} - 24546 T^{5} - 158105 T^{6} - 24546 p T^{7} + 7470 p^{2} T^{8} + 78 p^{3} T^{9} - 18 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
43 1−18T+114T2−682T3+7188T4−33492T5+63039T6−33492pT7+7188p2T8−682p3T9+114p4T10−18p5T11+p6T12 1 - 18 T + 114 T^{2} - 682 T^{3} + 7188 T^{4} - 33492 T^{5} + 63039 T^{6} - 33492 p T^{7} + 7188 p^{2} T^{8} - 682 p^{3} T^{9} + 114 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
47 (1−3T+117T2−309T3+117pT4−3p2T5+p3T6)2 ( 1 - 3 T + 117 T^{2} - 309 T^{3} + 117 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
53 1+9T−36T2−873T3−1179T4+26334T5+272077T6+26334pT7−1179p2T8−873p3T9−36p4T10+9p5T11+p6T12 1 + 9 T - 36 T^{2} - 873 T^{3} - 1179 T^{4} + 26334 T^{5} + 272077 T^{6} + 26334 p T^{7} - 1179 p^{2} T^{8} - 873 p^{3} T^{9} - 36 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}
59 (1−4T+76T2−11pT3+76pT4−4p2T5+p3T6)2 ( 1 - 4 T + 76 T^{2} - 11 p T^{3} + 76 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1+4T+48T2−229T3+48pT4+4p2T5+p3T6)2 ( 1 + 4 T + 48 T^{2} - 229 T^{3} + 48 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 (1+5T+143T2+521T3+143pT4+5p2T5+p3T6)2 ( 1 + 5 T + 143 T^{2} + 521 T^{3} + 143 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2}
71 (1+7T+163T2+895T3+163pT4+7p2T5+p3T6)2 ( 1 + 7 T + 163 T^{2} + 895 T^{3} + 163 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1+25T+254T2+2073T3+20533T4+115046T5+366817T6+115046pT7+20533p2T8+2073p3T9+254p4T10+25p5T11+p6T12 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 115046 p T^{7} + 20533 p^{2} T^{8} + 2073 p^{3} T^{9} + 254 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12}
79 (1+7T+93T2+335T3+93pT4+7p2T5+p3T6)2 ( 1 + 7 T + 93 T^{2} + 335 T^{3} + 93 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 1+8T−180T2−518T3+29404T4+32420T5−2713585T6+32420pT7+29404p2T8−518p3T9−180p4T10+8p5T11+p6T12 1 + 8 T - 180 T^{2} - 518 T^{3} + 29404 T^{4} + 32420 T^{5} - 2713585 T^{6} + 32420 p T^{7} + 29404 p^{2} T^{8} - 518 p^{3} T^{9} - 180 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
89 1−9T−180T2+729T3+31041T4−54846T5−2925911T6−54846pT7+31041p2T8+729p3T9−180p4T10−9p5T11+p6T12 1 - 9 T - 180 T^{2} + 729 T^{3} + 31041 T^{4} - 54846 T^{5} - 2925911 T^{6} - 54846 p T^{7} + 31041 p^{2} T^{8} + 729 p^{3} T^{9} - 180 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
97 1+28T+257T2+2820T3+59506T4+545924T5+3126001T6+545924pT7+59506p2T8+2820p3T9+257p4T10+28p5T11+p6T12 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 545924 p T^{7} + 59506 p^{2} T^{8} + 2820 p^{3} T^{9} + 257 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12}
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   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

−6.35267916410940766044169833535, −6.00702798748693717238636081135, −5.98877348263525930353623735776, −5.88147479707962295150086703306, −5.65919337235591364058034300382, −5.42014497308620620823936540920, −5.35586854944625954041454332849, −4.79589134688594176208177987037, −4.70626990288551518663047795372, −4.65111648849481329150113311472, −4.45702528389346279096867222059, −4.02227332237227663362504615260, −3.98063074580587535247293918254, −3.10764862115675558093630375663, −2.96157656912337613156547855326, −2.85955897377796404726866282930, −2.83395170584456424428951862227, −2.67522571548593585623230381822, −2.32773581164894388700557918850, −1.90745923851789441139476388489, −1.74795103451137140832114710543, −1.45791199851360867375119771818, −1.09855602244758995650226951663, −1.07986844095079761630771642186, −0.71931949668298495679503022046, 0.71931949668298495679503022046, 1.07986844095079761630771642186, 1.09855602244758995650226951663, 1.45791199851360867375119771818, 1.74795103451137140832114710543, 1.90745923851789441139476388489, 2.32773581164894388700557918850, 2.67522571548593585623230381822, 2.83395170584456424428951862227, 2.85955897377796404726866282930, 2.96157656912337613156547855326, 3.10764862115675558093630375663, 3.98063074580587535247293918254, 4.02227332237227663362504615260, 4.45702528389346279096867222059, 4.65111648849481329150113311472, 4.70626990288551518663047795372, 4.79589134688594176208177987037, 5.35586854944625954041454332849, 5.42014497308620620823936540920, 5.65919337235591364058034300382, 5.88147479707962295150086703306, 5.98877348263525930353623735776, 6.00702798748693717238636081135, 6.35267916410940766044169833535

Graph of the ZZ-function along the critical line

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