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Properties

Label 2-34e2-1156.375-c0-0-0
Degree 22
Conductor 11561156
Sign −0.701−0.712i-0.701 - 0.712i
Analytic cond. 0.5769190.576919
Root an. cond. 0.7595510.759551
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.273 + 0.961i)2-s + (−0.850 − 0.526i)4-s + (1.09 + 0.995i)5-s + (0.739 − 0.673i)8-s + (−0.982 − 0.183i)9-s + (−1.25 + 0.778i)10-s + (−1.45 + 1.32i)13-s + (0.445 + 0.895i)16-s + (0.445 + 0.895i)17-s + (0.445 − 0.895i)18-s + (−0.404 − 1.42i)20-s + (0.109 + 1.17i)25-s + (−0.876 − 1.75i)26-s + (1.67 + 1.03i)29-s + (−0.982 + 0.183i)32-s + ⋯
L(s)  = 1  + (−0.273 + 0.961i)2-s + (−0.850 − 0.526i)4-s + (1.09 + 0.995i)5-s + (0.739 − 0.673i)8-s + (−0.982 − 0.183i)9-s + (−1.25 + 0.778i)10-s + (−1.45 + 1.32i)13-s + (0.445 + 0.895i)16-s + (0.445 + 0.895i)17-s + (0.445 − 0.895i)18-s + (−0.404 − 1.42i)20-s + (0.109 + 1.17i)25-s + (−0.876 − 1.75i)26-s + (1.67 + 1.03i)29-s + (−0.982 + 0.183i)32-s + ⋯

Functional equation

Λ(s)=(1156s/2 ΓC(s) L(s)=((−0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1156s/2 ΓC(s) L(s)=((−0.701−0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: −0.701−0.712i-0.701 - 0.712i
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(375,⋅)\chi_{1156} (375, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1156, ( :0), −0.701−0.712i)(2,\ 1156,\ (\ :0),\ -0.701 - 0.712i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.85635958890.8563595889
L(12)L(\frac12) ≈\approx 0.85635958890.8563595889
L(1)L(1) 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+(0.273−0.961i)T 1 + (0.273 - 0.961i)T
17 1+(−0.445−0.895i)T 1 + (-0.445 - 0.895i)T
good3 1+(0.982+0.183i)T2 1 + (0.982 + 0.183i)T^{2}
5 1+(−1.09−0.995i)T+(0.0922+0.995i)T2 1 + (-1.09 - 0.995i)T + (0.0922 + 0.995i)T^{2}
7 1+(−0.932−0.361i)T2 1 + (-0.932 - 0.361i)T^{2}
11 1+(−0.445−0.895i)T2 1 + (-0.445 - 0.895i)T^{2}
13 1+(1.45−1.32i)T+(0.0922−0.995i)T2 1 + (1.45 - 1.32i)T + (0.0922 - 0.995i)T^{2}
19 1+(0.850−0.526i)T2 1 + (0.850 - 0.526i)T^{2}
23 1+(−0.932−0.361i)T2 1 + (-0.932 - 0.361i)T^{2}
29 1+(−1.67−1.03i)T+(0.445+0.895i)T2 1 + (-1.67 - 1.03i)T + (0.445 + 0.895i)T^{2}
31 1+(−0.0922+0.995i)T2 1 + (-0.0922 + 0.995i)T^{2}
37 1+(−0.329−0.436i)T+(−0.273+0.961i)T2 1 + (-0.329 - 0.436i)T + (-0.273 + 0.961i)T^{2}
41 1+(−0.136+1.47i)T+(−0.982−0.183i)T2 1 + (-0.136 + 1.47i)T + (-0.982 - 0.183i)T^{2}
43 1+(0.602+0.798i)T2 1 + (0.602 + 0.798i)T^{2}
47 1+(−0.932+0.361i)T2 1 + (-0.932 + 0.361i)T^{2}
53 1+(1.83+0.342i)T+(0.932+0.361i)T2 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2}
59 1+(−0.739+0.673i)T2 1 + (-0.739 + 0.673i)T^{2}
61 1+(−0.172−0.0666i)T+(0.739+0.673i)T2 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2}
67 1+(0.850−0.526i)T2 1 + (0.850 - 0.526i)T^{2}
71 1+(−0.932−0.361i)T2 1 + (-0.932 - 0.361i)T^{2}
73 1+(0.876+1.75i)T+(−0.602+0.798i)T2 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2}
79 1+(0.850−0.526i)T2 1 + (0.850 - 0.526i)T^{2}
83 1+(0.982−0.183i)T2 1 + (0.982 - 0.183i)T^{2}
89 1+(0.404+0.368i)T+(0.0922+0.995i)T2 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2}
97 1+(−1.67−0.312i)T+(0.932+0.361i)T2 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)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

−10.17498748189973755384957956010, −9.364369583629410239800649844667, −8.762883869714964361025263145182, −7.68733132265928789970097978944, −6.79451942785087842439763683375, −6.30961952265729104644506766670, −5.48672023804609392908304892759, −4.54236210323809808237790988735, −3.09388174612945160688824368725, −1.92383587300622588798351419297, 0.850730756124675275330105089148, 2.39415563935079843228643075734, 2.96715539179495410502403970997, 4.68181236645547792697953891712, 5.17272433818413914177593384793, 5.99957303580887622659680052670, 7.60392472927279573344576576042, 8.267931069885464755022162971854, 9.078436745206874386359748733090, 9.865036056923953572359616650425

Graph of the ZZ-function along the critical line