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Properties

Label 6-8512e3-1.1-c1e3-0-5
Degree 66
Conductor 616729673728616729673728
Sign −1-1
Analytic cond. 313997.313997.
Root an. cond. 8.244318.24431
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 3·7-s − 9-s − 3·11-s − 8·13-s + 2·15-s + 9·17-s − 3·19-s − 3·21-s − 4·23-s − 2·25-s + 5·27-s − 11·29-s + 5·31-s + 3·33-s − 6·35-s − 4·37-s + 8·39-s + 11·41-s − 4·43-s + 2·45-s + 2·47-s + 6·49-s − 9·51-s − 9·53-s + 6·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1.13·7-s − 1/3·9-s − 0.904·11-s − 2.21·13-s + 0.516·15-s + 2.18·17-s − 0.688·19-s − 0.654·21-s − 0.834·23-s − 2/5·25-s + 0.962·27-s − 2.04·29-s + 0.898·31-s + 0.522·33-s − 1.01·35-s − 0.657·37-s + 1.28·39-s + 1.71·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s + 6/7·49-s − 1.26·51-s − 1.23·53-s + 0.809·55-s + ⋯

Functional equation

Λ(s)=((218⋅73⋅193)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅73⋅193)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 218⋅73⋅1932^{18} \cdot 7^{3} \cdot 19^{3}
Sign: −1-1
Analytic conductor: 313997.313997.
Root analytic conductor: 8.244318.24431
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 218⋅73⋅193, ( :1/2,1/2,1/2), −1)(6,\ 2^{18} \cdot 7^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
7C1C_1 (1−T)3 ( 1 - T )^{3}
19C1C_1 (1+T)3 ( 1 + T )^{3}
good3S4×C2S_4\times C_2 1+T+2T2−2T3+2pT4+p2T5+p3T6 1 + T + 2 T^{2} - 2 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
5S4×C2S_4\times C_2 1+2T+6T2+6T3+6pT4+2p2T5+p3T6 1 + 2 T + 6 T^{2} + 6 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1+3T+26T2+46T3+26pT4+3p2T5+p3T6 1 + 3 T + 26 T^{2} + 46 T^{3} + 26 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1+8T+50T2+192T3+50pT4+8p2T5+p3T6 1 + 8 T + 50 T^{2} + 192 T^{3} + 50 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−9T+4pT2−292T3+4p2T4−9p2T5+p3T6 1 - 9 T + 4 p T^{2} - 292 T^{3} + 4 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+4T+64T2+180T3+64pT4+4p2T5+p3T6 1 + 4 T + 64 T^{2} + 180 T^{3} + 64 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1+11T+102T2+636T3+102pT4+11p2T5+p3T6 1 + 11 T + 102 T^{2} + 636 T^{3} + 102 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−5T+76T2−314T3+76pT4−5p2T5+p3T6 1 - 5 T + 76 T^{2} - 314 T^{3} + 76 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+4T+92T2+246T3+92pT4+4p2T5+p3T6 1 + 4 T + 92 T^{2} + 246 T^{3} + 92 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−11T+110T2−622T3+110pT4−11p2T5+p3T6 1 - 11 T + 110 T^{2} - 622 T^{3} + 110 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1+4T+37T2+376T3+37pT4+4p2T5+p3T6 1 + 4 T + 37 T^{2} + 376 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−2T+66T2−372T3+66pT4−2p2T5+p3T6 1 - 2 T + 66 T^{2} - 372 T^{3} + 66 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+9T+120T2+632T3+120pT4+9p2T5+p3T6 1 + 9 T + 120 T^{2} + 632 T^{3} + 120 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+12T+132T2+1308T3+132pT4+12p2T5+p3T6 1 + 12 T + 132 T^{2} + 1308 T^{3} + 132 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−2T+26T2−42T3+26pT4−2p2T5+p3T6 1 - 2 T + 26 T^{2} - 42 T^{3} + 26 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1+9T+218T2+1192T3+218pT4+9p2T5+p3T6 1 + 9 T + 218 T^{2} + 1192 T^{3} + 218 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+158T2+58T3+158pT4+p3T6 1 + 158 T^{2} + 58 T^{3} + 158 p T^{4} + p^{3} T^{6}
73S4×C2S_4\times C_2 1+21T+302T2+2764T3+302pT4+21p2T5+p3T6 1 + 21 T + 302 T^{2} + 2764 T^{3} + 302 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+6T+89T2−68T3+89pT4+6p2T5+p3T6 1 + 6 T + 89 T^{2} - 68 T^{3} + 89 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+7T+204T2+1062T3+204pT4+7p2T5+p3T6 1 + 7 T + 204 T^{2} + 1062 T^{3} + 204 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+18T+335T2+3268T3+335pT4+18p2T5+p3T6 1 + 18 T + 335 T^{2} + 3268 T^{3} + 335 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−28T+330T2−2896T3+330pT4−28p2T5+p3T6 1 - 28 T + 330 T^{2} - 2896 T^{3} + 330 p T^{4} - 28 p^{2} T^{5} + p^{3} 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

−7.34594761472028925664505815636, −7.32917433019601515780686071811, −6.98078304800888180086114323289, −6.32418828383105012292227726519, −6.06779315280294485683854622066, −6.05538046487009661587593475438, −6.00993816660285712310885482119, −5.45756285905518483488033473057, −5.30695977336905398355720072510, −5.14930638794445310424244747070, −4.97566284443671376326894139984, −4.54055392196831253313043191472, −4.52515640708979219378776444005, −4.12151404807576453470217569242, −4.10542585046753475281499223433, −3.63034305755035502871364709165, −3.15051257926425695128695552894, −3.10860984009089974773313640115, −2.96887781167001649968305514430, −2.36084623796459330576927140568, −2.28957320100449844668261461846, −1.96008575953867797219314147658, −1.55043815102810441727570080043, −1.23428361569132925743125456303, −0.882175080876133348166542961091, 0, 0, 0, 0.882175080876133348166542961091, 1.23428361569132925743125456303, 1.55043815102810441727570080043, 1.96008575953867797219314147658, 2.28957320100449844668261461846, 2.36084623796459330576927140568, 2.96887781167001649968305514430, 3.10860984009089974773313640115, 3.15051257926425695128695552894, 3.63034305755035502871364709165, 4.10542585046753475281499223433, 4.12151404807576453470217569242, 4.52515640708979219378776444005, 4.54055392196831253313043191472, 4.97566284443671376326894139984, 5.14930638794445310424244747070, 5.30695977336905398355720072510, 5.45756285905518483488033473057, 6.00993816660285712310885482119, 6.05538046487009661587593475438, 6.06779315280294485683854622066, 6.32418828383105012292227726519, 6.98078304800888180086114323289, 7.32917433019601515780686071811, 7.34594761472028925664505815636

Graph of the ZZ-function along the critical line