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Properties

Label 12-378e6-1.1-c1e6-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 8-s + 6·11-s − 3·13-s + 3·19-s + 6·23-s + 9·25-s − 9·27-s − 3·29-s − 3·31-s + 12·37-s − 3·40-s − 12·41-s − 6·43-s + 24·47-s + 18·55-s − 24·59-s − 3·61-s − 9·65-s + 3·67-s − 12·71-s + 3·73-s + 33·79-s − 6·88-s + 21·89-s + 9·95-s − 15·97-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.353·8-s + 1.80·11-s − 0.832·13-s + 0.688·19-s + 1.25·23-s + 9/5·25-s − 1.73·27-s − 0.557·29-s − 0.538·31-s + 1.97·37-s − 0.474·40-s − 1.87·41-s − 0.914·43-s + 3.50·47-s + 2.42·55-s − 3.12·59-s − 0.384·61-s − 1.11·65-s + 0.366·67-s − 1.42·71-s + 0.351·73-s + 3.71·79-s − 0.639·88-s + 2.22·89-s + 0.923·95-s − 1.52·97-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 4.7925729654.792572965
L(12)L(\frac12) ≈\approx 4.7925729654.792572965
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+T3+T6 1 + T^{3} + T^{6}
3 1+p2T3+p3T6 1 + p^{2} T^{3} + p^{3} T^{6}
7 1+T3+T6 1 + T^{3} + T^{6}
good5 1−3T+18T3−36T4−3p2T5+379T6−3p3T7−36p2T8+18p3T9−3p5T11+p6T12 1 - 3 T + 18 T^{3} - 36 T^{4} - 3 p^{2} T^{5} + 379 T^{6} - 3 p^{3} T^{7} - 36 p^{2} T^{8} + 18 p^{3} T^{9} - 3 p^{5} T^{11} + p^{6} T^{12}
11 1−6T+18T2−36T3−108T4+912T5−3203T6+912pT7−108p2T8−36p3T9+18p4T10−6p5T11+p6T12 1 - 6 T + 18 T^{2} - 36 T^{3} - 108 T^{4} + 912 T^{5} - 3203 T^{6} + 912 p T^{7} - 108 p^{2} T^{8} - 36 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
13 1+3T+6T2+8T3−27T4−891T5−3483T6−891pT7−27p2T8+8p3T9+6p4T10+3p5T11+p6T12 1 + 3 T + 6 T^{2} + 8 T^{3} - 27 T^{4} - 891 T^{5} - 3483 T^{6} - 891 p T^{7} - 27 p^{2} T^{8} + 8 p^{3} T^{9} + 6 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
17 1−42T2+18T3+1050T4−378T5−20081T6−378pT7+1050p2T8+18p3T9−42p4T10+p6T12 1 - 42 T^{2} + 18 T^{3} + 1050 T^{4} - 378 T^{5} - 20081 T^{6} - 378 p T^{7} + 1050 p^{2} T^{8} + 18 p^{3} T^{9} - 42 p^{4} T^{10} + p^{6} T^{12}
19 1−3T−24T2+23T3+279T4+666T5−6501T6+666pT7+279p2T8+23p3T9−24p4T10−3p5T11+p6T12 1 - 3 T - 24 T^{2} + 23 T^{3} + 279 T^{4} + 666 T^{5} - 6501 T^{6} + 666 p T^{7} + 279 p^{2} T^{8} + 23 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
23 1−6T−36T2+396T3−810T4−5874T5+51733T6−5874pT7−810p2T8+396p3T9−36p4T10−6p5T11+p6T12 1 - 6 T - 36 T^{2} + 396 T^{3} - 810 T^{4} - 5874 T^{5} + 51733 T^{6} - 5874 p T^{7} - 810 p^{2} T^{8} + 396 p^{3} T^{9} - 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
29 1+3T+108T2+90T3+4851T4−3507T5+149293T6−3507pT7+4851p2T8+90p3T9+108p4T10+3p5T11+p6T12 1 + 3 T + 108 T^{2} + 90 T^{3} + 4851 T^{4} - 3507 T^{5} + 149293 T^{6} - 3507 p T^{7} + 4851 p^{2} T^{8} + 90 p^{3} T^{9} + 108 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
31 1+3T+24T2+296T3+1530T4+7119T5+55665T6+7119pT7+1530p2T8+296p3T9+24p4T10+3p5T11+p6T12 1 + 3 T + 24 T^{2} + 296 T^{3} + 1530 T^{4} + 7119 T^{5} + 55665 T^{6} + 7119 p T^{7} + 1530 p^{2} T^{8} + 296 p^{3} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
37 1−12T+21T2+284T3+18T4−15444T5+118797T6−15444pT7+18p2T8+284p3T9+21p4T10−12p5T11+p6T12 1 - 12 T + 21 T^{2} + 284 T^{3} + 18 T^{4} - 15444 T^{5} + 118797 T^{6} - 15444 p T^{7} + 18 p^{2} T^{8} + 284 p^{3} T^{9} + 21 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
41 1+12T+144T2+1044T3+9216T4+58152T5+440263T6+58152pT7+9216p2T8+1044p3T9+144p4T10+12p5T11+p6T12 1 + 12 T + 144 T^{2} + 1044 T^{3} + 9216 T^{4} + 58152 T^{5} + 440263 T^{6} + 58152 p T^{7} + 9216 p^{2} T^{8} + 1044 p^{3} T^{9} + 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}
43 1+6T−6T2+284T3−684T4−7920T5+123837T6−7920pT7−684p2T8+284p3T9−6p4T10+6p5T11+p6T12 1 + 6 T - 6 T^{2} + 284 T^{3} - 684 T^{4} - 7920 T^{5} + 123837 T^{6} - 7920 p T^{7} - 684 p^{2} T^{8} + 284 p^{3} T^{9} - 6 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
47 1−24T+306T2−2844T3+25254T4−215106T5+1621333T6−215106pT7+25254p2T8−2844p3T9+306p4T10−24p5T11+p6T12 1 - 24 T + 306 T^{2} - 2844 T^{3} + 25254 T^{4} - 215106 T^{5} + 1621333 T^{6} - 215106 p T^{7} + 25254 p^{2} T^{8} - 2844 p^{3} T^{9} + 306 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12}
53 (1+42T2+153T3+42pT4+p3T6)2 ( 1 + 42 T^{2} + 153 T^{3} + 42 p T^{4} + p^{3} T^{6} )^{2}
59 1+24T+252T2+1395T3−2475T4−150699T5−1633823T6−150699pT7−2475p2T8+1395p3T9+252p4T10+24p5T11+p6T12 1 + 24 T + 252 T^{2} + 1395 T^{3} - 2475 T^{4} - 150699 T^{5} - 1633823 T^{6} - 150699 p T^{7} - 2475 p^{2} T^{8} + 1395 p^{3} T^{9} + 252 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12}
61 1+3T+114T2+926T3+12897T4+70029T5+1006281T6+70029pT7+12897p2T8+926p3T9+114p4T10+3p5T11+p6T12 1 + 3 T + 114 T^{2} + 926 T^{3} + 12897 T^{4} + 70029 T^{5} + 1006281 T^{6} + 70029 p T^{7} + 12897 p^{2} T^{8} + 926 p^{3} T^{9} + 114 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
67 1−3T−96T2+1094T3−3735T4−48879T5+940191T6−48879pT7−3735p2T8+1094p3T9−96p4T10−3p5T11+p6T12 1 - 3 T - 96 T^{2} + 1094 T^{3} - 3735 T^{4} - 48879 T^{5} + 940191 T^{6} - 48879 p T^{7} - 3735 p^{2} T^{8} + 1094 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
71 1+12T−24T2−738T3−228T4+5556T5−117857T6+5556pT7−228p2T8−738p3T9−24p4T10+12p5T11+p6T12 1 + 12 T - 24 T^{2} - 738 T^{3} - 228 T^{4} + 5556 T^{5} - 117857 T^{6} + 5556 p T^{7} - 228 p^{2} T^{8} - 738 p^{3} T^{9} - 24 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12}
73 1−3T−96T2+635T3+1935T4−19872T5+150873T6−19872pT7+1935p2T8+635p3T9−96p4T10−3p5T11+p6T12 1 - 3 T - 96 T^{2} + 635 T^{3} + 1935 T^{4} - 19872 T^{5} + 150873 T^{6} - 19872 p T^{7} + 1935 p^{2} T^{8} + 635 p^{3} T^{9} - 96 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
79 1−33T+492T2−4402T3+17532T4+134901T5−2452347T6+134901pT7+17532p2T8−4402p3T9+492p4T10−33p5T11+p6T12 1 - 33 T + 492 T^{2} - 4402 T^{3} + 17532 T^{4} + 134901 T^{5} - 2452347 T^{6} + 134901 p T^{7} + 17532 p^{2} T^{8} - 4402 p^{3} T^{9} + 492 p^{4} T^{10} - 33 p^{5} T^{11} + p^{6} T^{12}
83 1+144T2+774T3+252pT4+72810T5+2086741T6+72810pT7+252p3T8+774p3T9+144p4T10+p6T12 1 + 144 T^{2} + 774 T^{3} + 252 p T^{4} + 72810 T^{5} + 2086741 T^{6} + 72810 p T^{7} + 252 p^{3} T^{8} + 774 p^{3} T^{9} + 144 p^{4} T^{10} + p^{6} T^{12}
89 1−21T+138T2+891T3−11325T4−132708T5+2900905T6−132708pT7−11325p2T8+891p3T9+138p4T10−21p5T11+p6T12 1 - 21 T + 138 T^{2} + 891 T^{3} - 11325 T^{4} - 132708 T^{5} + 2900905 T^{6} - 132708 p T^{7} - 11325 p^{2} T^{8} + 891 p^{3} T^{9} + 138 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12}
97 1+15T+120T2+2138T3+14679T4+127917T5+2485869T6+127917pT7+14679p2T8+2138p3T9+120p4T10+15p5T11+p6T12 1 + 15 T + 120 T^{2} + 2138 T^{3} + 14679 T^{4} + 127917 T^{5} + 2485869 T^{6} + 127917 p T^{7} + 14679 p^{2} T^{8} + 2138 p^{3} T^{9} + 120 p^{4} T^{10} + 15 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

−6.19469640954300286742955724894, −5.89938642267690145183335393344, −5.88689647891834501647123125427, −5.77420506695725453505405418021, −5.33353953963057264739781476701, −5.33058048778540071897575458397, −5.25446407535737281669262475341, −4.88112144170358287409169555860, −4.48056608614308113383340474784, −4.44483479195359510960871454432, −4.42752731918502751554233371761, −4.42246261116115003567997277719, −3.63334277362969873672709770248, −3.59192359598209276680384063153, −3.51306862946926595578816879310, −3.16563681769817311945100985196, −3.00748067489457519661814878869, −2.98565728353735754904141128032, −2.25936344599167431064747569560, −2.13104643515048270328639315970, −1.99239890228082090450145361907, −1.88344327181844075184732071251, −1.36587140397128264326838243014, −0.837443845706437985596845036752, −0.75551355626911789136133263968, 0.75551355626911789136133263968, 0.837443845706437985596845036752, 1.36587140397128264326838243014, 1.88344327181844075184732071251, 1.99239890228082090450145361907, 2.13104643515048270328639315970, 2.25936344599167431064747569560, 2.98565728353735754904141128032, 3.00748067489457519661814878869, 3.16563681769817311945100985196, 3.51306862946926595578816879310, 3.59192359598209276680384063153, 3.63334277362969873672709770248, 4.42246261116115003567997277719, 4.42752731918502751554233371761, 4.44483479195359510960871454432, 4.48056608614308113383340474784, 4.88112144170358287409169555860, 5.25446407535737281669262475341, 5.33058048778540071897575458397, 5.33353953963057264739781476701, 5.77420506695725453505405418021, 5.88689647891834501647123125427, 5.89938642267690145183335393344, 6.19469640954300286742955724894

Graph of the ZZ-function along the critical line

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