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Properties

Label 2-34e2-1156.171-c0-0-0
Degree 22
Conductor 11561156
Sign 0.887+0.460i0.887 + 0.460i
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  + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (0.0170 + 0.183i)5-s + (0.0922 − 0.995i)8-s + (0.932 + 0.361i)9-s + (0.0822 − 0.165i)10-s + (0.172 − 1.85i)13-s + (−0.602 + 0.798i)16-s + (−0.602 + 0.798i)17-s + (−0.602 − 0.798i)18-s + (−0.156 + 0.0971i)20-s + (0.949 − 0.177i)25-s + (−1.12 + 1.48i)26-s + (0.831 + 1.66i)29-s + (0.932 − 0.361i)32-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)2-s + (0.445 + 0.895i)4-s + (0.0170 + 0.183i)5-s + (0.0922 − 0.995i)8-s + (0.932 + 0.361i)9-s + (0.0822 − 0.165i)10-s + (0.172 − 1.85i)13-s + (−0.602 + 0.798i)16-s + (−0.602 + 0.798i)17-s + (−0.602 − 0.798i)18-s + (−0.156 + 0.0971i)20-s + (0.949 − 0.177i)25-s + (−1.12 + 1.48i)26-s + (0.831 + 1.66i)29-s + (0.932 − 0.361i)32-s + ⋯

Functional equation

Λ(s)=(1156s/2 ΓC(s) L(s)=((0.887+0.460i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1156s/2 ΓC(s) L(s)=((0.887+0.460i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: 0.887+0.460i0.887 + 0.460i
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(171,⋅)\chi_{1156} (171, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1156, ( :0), 0.887+0.460i)(2,\ 1156,\ (\ :0),\ 0.887 + 0.460i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.77235424550.7723542455
L(12)L(\frac12) ≈\approx 0.77235424550.7723542455
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+(0.850+0.526i)T 1 + (0.850 + 0.526i)T
17 1+(0.602−0.798i)T 1 + (0.602 - 0.798i)T
good3 1+(−0.932−0.361i)T2 1 + (-0.932 - 0.361i)T^{2}
5 1+(−0.0170−0.183i)T+(−0.982+0.183i)T2 1 + (-0.0170 - 0.183i)T + (-0.982 + 0.183i)T^{2}
7 1+(−0.739−0.673i)T2 1 + (-0.739 - 0.673i)T^{2}
11 1+(0.602−0.798i)T2 1 + (0.602 - 0.798i)T^{2}
13 1+(−0.172+1.85i)T+(−0.982−0.183i)T2 1 + (-0.172 + 1.85i)T + (-0.982 - 0.183i)T^{2}
19 1+(−0.445+0.895i)T2 1 + (-0.445 + 0.895i)T^{2}
23 1+(−0.739−0.673i)T2 1 + (-0.739 - 0.673i)T^{2}
29 1+(−0.831−1.66i)T+(−0.602+0.798i)T2 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2}
31 1+(0.982+0.183i)T2 1 + (0.982 + 0.183i)T^{2}
37 1+(−0.465+1.63i)T+(−0.850−0.526i)T2 1 + (-0.465 + 1.63i)T + (-0.850 - 0.526i)T^{2}
41 1+(0.181+0.0339i)T+(0.932+0.361i)T2 1 + (0.181 + 0.0339i)T + (0.932 + 0.361i)T^{2}
43 1+(0.273−0.961i)T2 1 + (0.273 - 0.961i)T^{2}
47 1+(−0.739+0.673i)T2 1 + (-0.739 + 0.673i)T^{2}
53 1+(−1.37−0.533i)T+(0.739+0.673i)T2 1 + (-1.37 - 0.533i)T + (0.739 + 0.673i)T^{2}
59 1+(−0.0922+0.995i)T2 1 + (-0.0922 + 0.995i)T^{2}
61 1+(1.45+1.32i)T+(0.0922+0.995i)T2 1 + (1.45 + 1.32i)T + (0.0922 + 0.995i)T^{2}
67 1+(−0.445+0.895i)T2 1 + (-0.445 + 0.895i)T^{2}
71 1+(−0.739−0.673i)T2 1 + (-0.739 - 0.673i)T^{2}
73 1+(1.12−1.48i)T+(−0.273−0.961i)T2 1 + (1.12 - 1.48i)T + (-0.273 - 0.961i)T^{2}
79 1+(−0.445+0.895i)T2 1 + (-0.445 + 0.895i)T^{2}
83 1+(−0.932+0.361i)T2 1 + (-0.932 + 0.361i)T^{2}
89 1+(0.156+1.69i)T+(−0.982+0.183i)T2 1 + (0.156 + 1.69i)T + (-0.982 + 0.183i)T^{2}
97 1+(−0.831−0.322i)T+(0.739+0.673i)T2 1 + (-0.831 - 0.322i)T + (0.739 + 0.673i)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

−10.25015994314784665826251052627, −9.073859945789747553181036171205, −8.412976924467154628502987840882, −7.56939769053266226901422825568, −6.93236944455164994559250706208, −5.82128572419878965221904714830, −4.58152985324728172748177021756, −3.49192391556492926419752601929, −2.53128946298729961469836672404, −1.19409123053474964611097160587, 1.26359413014448654490991500921, 2.45907468546747568195689857256, 4.22189945014423448607195130546, 4.88647119347948135699681688565, 6.28966686511274309968037154781, 6.76476919813235285430692352468, 7.50242379020442123633830183159, 8.595712553922845983825962680516, 9.183188401365288943121163420154, 9.833966130512109272327170612822

Graph of the ZZ-function along the critical line