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Properties

Label 2-34e2-1156.203-c0-0-0
Degree 22
Conductor 11561156
Sign 0.701+0.712i0.701 + 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 + (−0.907 + 0.995i)5-s + (−0.739 + 0.673i)8-s + (0.982 + 0.183i)9-s + (0.709 + 1.14i)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 + (1.29 − 0.368i)20-s + (−0.0752 − 0.811i)25-s + (−0.876 − 1.75i)26-s + (−0.193 + 0.312i)29-s + (0.982 − 0.183i)32-s + ⋯
L(s)  = 1  + (0.273 − 0.961i)2-s + (−0.850 − 0.526i)4-s + (−0.907 + 0.995i)5-s + (−0.739 + 0.673i)8-s + (0.982 + 0.183i)9-s + (0.709 + 1.14i)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 + (1.29 − 0.368i)20-s + (−0.0752 − 0.811i)25-s + (−0.876 − 1.75i)26-s + (−0.193 + 0.312i)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+0.712i0.701 + 0.712i
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(203,⋅)\chi_{1156} (203, \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 1.0521496481.052149648
L(12)L(\frac12) ≈\approx 1.0521496481.052149648
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+(0.907−0.995i)T+(−0.0922−0.995i)T2 1 + (0.907 - 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+(0.193−0.312i)T+(−0.445−0.895i)T2 1 + (0.193 - 0.312i)T + (-0.445 - 0.895i)T^{2}
31 1+(0.0922−0.995i)T2 1 + (0.0922 - 0.995i)T^{2}
37 1+(−1.53+1.15i)T+(0.273−0.961i)T2 1 + (-1.53 + 1.15i)T + (0.273 - 0.961i)T^{2}
41 1+(−1.34−0.124i)T+(0.982+0.183i)T2 1 + (-1.34 - 0.124i)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+(0.719−1.85i)T+(−0.739−0.673i)T2 1 + (0.719 - 1.85i)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+(0.328−0.163i)T+(0.602−0.798i)T2 1 + (0.328 - 0.163i)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+(0.193−1.03i)T+(−0.932−0.361i)T2 1 + (0.193 - 1.03i)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

−10.23151467611794046620350357169, −9.297342454502210641804252666874, −8.128671516944279732088370523676, −7.67895382535430247908202225206, −6.40697761167907435606660948107, −5.60404846513164993326201516111, −4.25126074267269618165755246343, −3.66086632386963976420986757119, −2.83338690964142211479623405222, −1.29975557571345099482002491333, 1.18408560457446018734734438584, 3.41870578512578101064778384167, 4.37391512214336856915205362607, 4.67267238947128959082283325837, 6.02882843987016664324519371206, 6.74612184402807536765117291850, 7.73783147409510055098761743305, 8.209342535383463263693337673970, 9.280091241294008570554254280553, 9.519480735128088465321731150082

Graph of the ZZ-function along the critical line