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Properties

Label 2-62-1.1-c1-0-2
Degree 22
Conductor 6262
Sign 11
Analytic cond. 0.4950720.495072
Root an. cond. 0.7036130.703613
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s + 8-s − 3·9-s − 2·10-s + 2·13-s + 16-s − 6·17-s − 3·18-s + 4·19-s − 2·20-s + 8·23-s − 25-s + 2·26-s + 2·29-s − 31-s + 32-s − 6·34-s − 3·36-s + 10·37-s + 4·38-s − 2·40-s − 6·41-s + 8·43-s + 6·45-s + 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 9-s − 0.632·10-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.371·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s − 0.316·40-s − 0.937·41-s + 1.21·43-s + 0.894·45-s + 1.17·46-s + ⋯

Functional equation

Λ(s)=(62s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(62s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6262    =    2⋅312 \cdot 31
Sign: 11
Analytic conductor: 0.4950720.495072
Root analytic conductor: 0.7036130.703613
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 62, ( :1/2), 1)(2,\ 62,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1087406601.108740660
L(12)L(\frac12) ≈\approx 1.1087406601.108740660
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 1 - T
31 1+T 1 + T
good3 1+pT2 1 + p T^{2}
5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 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+12T+pT2 1 + 12 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+12T+pT2 1 + 12 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+8T+pT2 1 + 8 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 1−2T+pT2 1 - 2 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

−15.05691750776431447573088356666, −13.90986476156025600386975635688, −12.88268162615932246882266504036, −11.52935425193863190686584011034, −11.06274569152331522479839207162, −9.056498267458590117851162621111, −7.76857292337562945056886541715, −6.32858579420662158347624019611, −4.74203912193695697715499891246, −3.16188554299090604324575557253, 3.16188554299090604324575557253, 4.74203912193695697715499891246, 6.32858579420662158347624019611, 7.76857292337562945056886541715, 9.056498267458590117851162621111, 11.06274569152331522479839207162, 11.52935425193863190686584011034, 12.88268162615932246882266504036, 13.90986476156025600386975635688, 15.05691750776431447573088356666

Graph of the ZZ-function along the critical line