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Properties

Label 6-825e3-1.1-c5e3-0-3
Degree 66
Conductor 561515625561515625
Sign −1-1
Analytic cond. 2.31655×1062.31655\times 10^{6}
Root an. cond. 11.502811.5028
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·2-s − 27·3-s − 11·4-s − 189·6-s + 172·7-s − 189·8-s + 486·9-s − 363·11-s + 297·12-s + 654·13-s + 1.20e3·14-s − 427·16-s + 2.36e3·17-s + 3.40e3·18-s − 2.87e3·19-s − 4.64e3·21-s − 2.54e3·22-s − 2.27e3·23-s + 5.10e3·24-s + 4.57e3·26-s − 7.29e3·27-s − 1.89e3·28-s − 7.73e3·29-s + 568·31-s − 1.30e3·32-s + 9.80e3·33-s + 1.65e4·34-s + ⋯
L(s)  = 1  + 1.23·2-s − 1.73·3-s − 0.343·4-s − 2.14·6-s + 1.32·7-s − 1.04·8-s + 2·9-s − 0.904·11-s + 0.595·12-s + 1.07·13-s + 1.64·14-s − 0.416·16-s + 1.98·17-s + 2.47·18-s − 1.82·19-s − 2.29·21-s − 1.11·22-s − 0.895·23-s + 1.80·24-s + 1.32·26-s − 1.92·27-s − 0.456·28-s − 1.70·29-s + 0.106·31-s − 0.225·32-s + 1.56·33-s + 2.45·34-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅113)s/2 ΓC(s)3 L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(6-s)\end{aligned}
Λ(s)=((33⋅56⋅113)s/2 ΓC(s+5/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅1133^{3} \cdot 5^{6} \cdot 11^{3}
Sign: −1-1
Analytic conductor: 2.31655×1062.31655\times 10^{6}
Root analytic conductor: 11.502811.5028
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 33⋅56⋅113, ( :5/2,5/2,5/2), −1)(6,\ 3^{3} \cdot 5^{6} \cdot 11^{3} ,\ ( \ : 5/2, 5/2, 5/2 ),\ -1 )

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 (1+p2T)3 ( 1 + p^{2} T )^{3}
5 1 1
11C1C_1 (1+p2T)3 ( 1 + p^{2} T )^{3}
good2S4×C2S_4\times C_2 1−7T+15p2T2−77p2T3+15p7T4−7p10T5+p15T6 1 - 7 T + 15 p^{2} T^{2} - 77 p^{2} T^{3} + 15 p^{7} T^{4} - 7 p^{10} T^{5} + p^{15} T^{6}
7S4×C2S_4\times C_2 1−172T+53317T2−5399144T3+53317p5T4−172p10T5+p15T6 1 - 172 T + 53317 T^{2} - 5399144 T^{3} + 53317 p^{5} T^{4} - 172 p^{10} T^{5} + p^{15} T^{6}
13S4×C2S_4\times C_2 1−654T+555091T2−167732852T3+555091p5T4−654p10T5+p15T6 1 - 654 T + 555091 T^{2} - 167732852 T^{3} + 555091 p^{5} T^{4} - 654 p^{10} T^{5} + p^{15} T^{6}
17S4×C2S_4\times C_2 1−2366T+4754783T2−6580937060T3+4754783p5T4−2366p10T5+p15T6 1 - 2366 T + 4754783 T^{2} - 6580937060 T^{3} + 4754783 p^{5} T^{4} - 2366 p^{10} T^{5} + p^{15} T^{6}
19S4×C2S_4\times C_2 1+2872T+9249705T2+14234256272T3+9249705p5T4+2872p10T5+p15T6 1 + 2872 T + 9249705 T^{2} + 14234256272 T^{3} + 9249705 p^{5} T^{4} + 2872 p^{10} T^{5} + p^{15} T^{6}
23S4×C2S_4\times C_2 1+2272T+18049957T2+25539838016T3+18049957p5T4+2272p10T5+p15T6 1 + 2272 T + 18049957 T^{2} + 25539838016 T^{3} + 18049957 p^{5} T^{4} + 2272 p^{10} T^{5} + p^{15} T^{6}
29S4×C2S_4\times C_2 1+7738T+75886547T2+322651167772T3+75886547p5T4+7738p10T5+p15T6 1 + 7738 T + 75886547 T^{2} + 322651167772 T^{3} + 75886547 p^{5} T^{4} + 7738 p^{10} T^{5} + p^{15} T^{6}
31S4×C2S_4\times C_2 1−568T+83955741T2−32812503440T3+83955741p5T4−568p10T5+p15T6 1 - 568 T + 83955741 T^{2} - 32812503440 T^{3} + 83955741 p^{5} T^{4} - 568 p^{10} T^{5} + p^{15} T^{6}
37S4×C2S_4\times C_2 1−9126T+153307915T2−1135693394116T3+153307915p5T4−9126p10T5+p15T6 1 - 9126 T + 153307915 T^{2} - 1135693394116 T^{3} + 153307915 p^{5} T^{4} - 9126 p^{10} T^{5} + p^{15} T^{6}
41S4×C2S_4\times C_2 1+8758T+117252903T2+906684659284T3+117252903p5T4+8758p10T5+p15T6 1 + 8758 T + 117252903 T^{2} + 906684659284 T^{3} + 117252903 p^{5} T^{4} + 8758 p^{10} T^{5} + p^{15} T^{6}
43S4×C2S_4\times C_2 1−14672T+370715025T2−3794906879008T3+370715025p5T4−14672p10T5+p15T6 1 - 14672 T + 370715025 T^{2} - 3794906879008 T^{3} + 370715025 p^{5} T^{4} - 14672 p^{10} T^{5} + p^{15} T^{6}
47S4×C2S_4\times C_2 1−19392T+652921165T2−7386008220288T3+652921165p5T4−19392p10T5+p15T6 1 - 19392 T + 652921165 T^{2} - 7386008220288 T^{3} + 652921165 p^{5} T^{4} - 19392 p^{10} T^{5} + p^{15} T^{6}
53S4×C2S_4\times C_2 1−4598T+900328507T2−4574622258916T3+900328507p5T4−4598p10T5+p15T6 1 - 4598 T + 900328507 T^{2} - 4574622258916 T^{3} + 900328507 p^{5} T^{4} - 4598 p^{10} T^{5} + p^{15} T^{6}
59S4×C2S_4\times C_2 1+9348T+1646289553T2+7098388384024T3+1646289553p5T4+9348p10T5+p15T6 1 + 9348 T + 1646289553 T^{2} + 7098388384024 T^{3} + 1646289553 p^{5} T^{4} + 9348 p^{10} T^{5} + p^{15} T^{6}
61S4×C2S_4\times C_2 1+60078T+49584271pT2+87982416745556T3+49584271p6T4+60078p10T5+p15T6 1 + 60078 T + 49584271 p T^{2} + 87982416745556 T^{3} + 49584271 p^{6} T^{4} + 60078 p^{10} T^{5} + p^{15} T^{6}
67S4×C2S_4\times C_2 1−38468T+3866400905T2−95393272971992T3+3866400905p5T4−38468p10T5+p15T6 1 - 38468 T + 3866400905 T^{2} - 95393272971992 T^{3} + 3866400905 p^{5} T^{4} - 38468 p^{10} T^{5} + p^{15} T^{6}
71S4×C2S_4\times C_2 1+74032T+6098518645T2+250129423986848T3+6098518645p5T4+74032p10T5+p15T6 1 + 74032 T + 6098518645 T^{2} + 250129423986848 T^{3} + 6098518645 p^{5} T^{4} + 74032 p^{10} T^{5} + p^{15} T^{6}
73S4×C2S_4\times C_2 1−44442T+6331091479T2−174512379795884T3+6331091479p5T4−44442p10T5+p15T6 1 - 44442 T + 6331091479 T^{2} - 174512379795884 T^{3} + 6331091479 p^{5} T^{4} - 44442 p^{10} T^{5} + p^{15} T^{6}
79S4×C2S_4\times C_2 1+108116T+11158675133T2+8537056071080pT3+11158675133p5T4+108116p10T5+p15T6 1 + 108116 T + 11158675133 T^{2} + 8537056071080 p T^{3} + 11158675133 p^{5} T^{4} + 108116 p^{10} T^{5} + p^{15} T^{6}
83S4×C2S_4\times C_2 1−81892T+2009956905T2+94678226672552T3+2009956905p5T4−81892p10T5+p15T6 1 - 81892 T + 2009956905 T^{2} + 94678226672552 T^{3} + 2009956905 p^{5} T^{4} - 81892 p^{10} T^{5} + p^{15} T^{6}
89S4×C2S_4\times C_2 1−167342T+25837929495T2−2027809825205668T3+25837929495p5T4−167342p10T5+p15T6 1 - 167342 T + 25837929495 T^{2} - 2027809825205668 T^{3} + 25837929495 p^{5} T^{4} - 167342 p^{10} T^{5} + p^{15} T^{6}
97S4×C2S_4\times C_2 1+159702T+28145564719T2+2389832506953716T3+28145564719p5T4+159702p10T5+p15T6 1 + 159702 T + 28145564719 T^{2} + 2389832506953716 T^{3} + 28145564719 p^{5} T^{4} + 159702 p^{10} T^{5} + p^{15} T^{6}
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   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.923843061495281321639171255843, −8.323033958376440858631661080611, −8.001767243415159885732918871340, −7.914330285723199381888454815096, −7.56837908225641648548646355406, −7.55712335077672776302365328732, −6.76116137485126518151913681885, −6.60856137316361275890159419583, −6.17957841290830233119713451007, −6.02916301198691018755960297134, −5.56116819804269304654440133871, −5.42747433102649607331012164044, −5.40474164098502051732765924007, −4.73376820302772111932277171051, −4.68436205643931953604344196178, −4.45973700874466101902621073428, −3.86024340195301388845383836803, −3.81257051325707501448787147813, −3.66114606219011617983742640275, −2.78456789897390155965773226999, −2.56557075570168019424297531847, −1.91340040158909911062276212887, −1.58627086240070032123947857490, −1.10079465515973933975113574052, −1.09325899895399695595104985822, 0, 0, 0, 1.09325899895399695595104985822, 1.10079465515973933975113574052, 1.58627086240070032123947857490, 1.91340040158909911062276212887, 2.56557075570168019424297531847, 2.78456789897390155965773226999, 3.66114606219011617983742640275, 3.81257051325707501448787147813, 3.86024340195301388845383836803, 4.45973700874466101902621073428, 4.68436205643931953604344196178, 4.73376820302772111932277171051, 5.40474164098502051732765924007, 5.42747433102649607331012164044, 5.56116819804269304654440133871, 6.02916301198691018755960297134, 6.17957841290830233119713451007, 6.60856137316361275890159419583, 6.76116137485126518151913681885, 7.55712335077672776302365328732, 7.56837908225641648548646355406, 7.914330285723199381888454815096, 8.001767243415159885732918871340, 8.323033958376440858631661080611, 8.923843061495281321639171255843

Graph of the ZZ-function along the critical line