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Properties

Label 2-56e2-1.1-c1-0-62
Degree 22
Conductor 31363136
Sign −1-1
Analytic cond. 25.041025.0410
Root an. cond. 5.004105.00410
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 3·9-s − 4·11-s + 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s + 2·37-s − 2·41-s − 4·43-s − 6·45-s − 8·47-s − 6·53-s − 8·55-s − 6·61-s + 4·65-s − 4·67-s + 8·71-s − 10·73-s − 16·79-s + 9·81-s − 8·83-s + 12·85-s + 6·89-s − 16·95-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s − 0.824·53-s − 1.07·55-s − 0.768·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 81-s − 0.878·83-s + 1.30·85-s + 0.635·89-s − 1.64·95-s + ⋯

Functional equation

Λ(s)=(3136s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3136s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31363136    =    26⋅722^{6} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 25.041025.0410
Root analytic conductor: 5.004105.00410
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3136, ( :1/2), −1)(2,\ 3136,\ (\ :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}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
good3 1+pT2 1 + p T^{2}
5 1−2T+pT2 1 - 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−6T+pT2 1 - 6 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.227785865636410870393529644581, −7.84054312303988042972335609462, −6.59264336057149769206055975154, −5.91740132798549001211232285063, −5.47493819124631815597461836959, −4.53211800578457486655010876487, −3.32773603072291404011890923635, −2.58182243115007516792385368988, −1.62407385510972851799092712087, 0, 1.62407385510972851799092712087, 2.58182243115007516792385368988, 3.32773603072291404011890923635, 4.53211800578457486655010876487, 5.47493819124631815597461836959, 5.91740132798549001211232285063, 6.59264336057149769206055975154, 7.84054312303988042972335609462, 8.227785865636410870393529644581

Graph of the ZZ-function along the critical line