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Properties

Label 2-34e2-1156.1055-c0-0-0
Degree 22
Conductor 11561156
Sign 0.781+0.624i0.781 + 0.624i
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  + (−0.602 − 0.798i)2-s + (−0.273 + 0.961i)4-s + (1.73 + 0.673i)5-s + (0.932 − 0.361i)8-s + (0.0922 − 0.995i)9-s + (−0.510 − 1.79i)10-s + (0.172 − 0.0666i)13-s + (−0.850 − 0.526i)16-s + (−0.850 − 0.526i)17-s + (−0.850 + 0.526i)18-s + (−1.12 + 1.48i)20-s + (1.83 + 1.66i)25-s + (−0.156 − 0.0971i)26-s + (−0.0505 + 0.177i)29-s + (0.0922 + 0.995i)32-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (−0.273 + 0.961i)4-s + (1.73 + 0.673i)5-s + (0.932 − 0.361i)8-s + (0.0922 − 0.995i)9-s + (−0.510 − 1.79i)10-s + (0.172 − 0.0666i)13-s + (−0.850 − 0.526i)16-s + (−0.850 − 0.526i)17-s + (−0.850 + 0.526i)18-s + (−1.12 + 1.48i)20-s + (1.83 + 1.66i)25-s + (−0.156 − 0.0971i)26-s + (−0.0505 + 0.177i)29-s + (0.0922 + 0.995i)32-s + ⋯

Functional equation

Λ(s)=(1156s/2 ΓC(s) L(s)=((0.781+0.624i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1156s/2 ΓC(s) L(s)=((0.781+0.624i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: 0.781+0.624i0.781 + 0.624i
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(1055,⋅)\chi_{1156} (1055, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1156, ( :0), 0.781+0.624i)(2,\ 1156,\ (\ :0),\ 0.781 + 0.624i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0184979891.018497989
L(12)L(\frac12) ≈\approx 1.0184979891.018497989
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+(0.602+0.798i)T 1 + (0.602 + 0.798i)T
17 1+(0.850+0.526i)T 1 + (0.850 + 0.526i)T
good3 1+(−0.0922+0.995i)T2 1 + (-0.0922 + 0.995i)T^{2}
5 1+(−1.73−0.673i)T+(0.739+0.673i)T2 1 + (-1.73 - 0.673i)T + (0.739 + 0.673i)T^{2}
7 1+(0.982+0.183i)T2 1 + (0.982 + 0.183i)T^{2}
11 1+(0.850+0.526i)T2 1 + (0.850 + 0.526i)T^{2}
13 1+(−0.172+0.0666i)T+(0.739−0.673i)T2 1 + (-0.172 + 0.0666i)T + (0.739 - 0.673i)T^{2}
19 1+(0.273+0.961i)T2 1 + (0.273 + 0.961i)T^{2}
23 1+(0.982+0.183i)T2 1 + (0.982 + 0.183i)T^{2}
29 1+(0.0505−0.177i)T+(−0.850−0.526i)T2 1 + (0.0505 - 0.177i)T + (-0.850 - 0.526i)T^{2}
31 1+(−0.739+0.673i)T2 1 + (-0.739 + 0.673i)T^{2}
37 1+(0.537−1.07i)T+(−0.602−0.798i)T2 1 + (0.537 - 1.07i)T + (-0.602 - 0.798i)T^{2}
41 1+(−1.37+1.25i)T+(0.0922−0.995i)T2 1 + (-1.37 + 1.25i)T + (0.0922 - 0.995i)T^{2}
43 1+(−0.445+0.895i)T2 1 + (-0.445 + 0.895i)T^{2}
47 1+(0.982−0.183i)T2 1 + (0.982 - 0.183i)T^{2}
53 1+(0.181−1.95i)T+(−0.982−0.183i)T2 1 + (0.181 - 1.95i)T + (-0.982 - 0.183i)T^{2}
59 1+(−0.932+0.361i)T2 1 + (-0.932 + 0.361i)T^{2}
61 1+(1.45+0.271i)T+(0.932+0.361i)T2 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2}
67 1+(0.273+0.961i)T2 1 + (0.273 + 0.961i)T^{2}
71 1+(0.982+0.183i)T2 1 + (0.982 + 0.183i)T^{2}
73 1+(0.156+0.0971i)T+(0.445+0.895i)T2 1 + (0.156 + 0.0971i)T + (0.445 + 0.895i)T^{2}
79 1+(0.273+0.961i)T2 1 + (0.273 + 0.961i)T^{2}
83 1+(−0.0922−0.995i)T2 1 + (-0.0922 - 0.995i)T^{2}
89 1+(1.12+0.435i)T+(0.739+0.673i)T2 1 + (1.12 + 0.435i)T + (0.739 + 0.673i)T^{2}
97 1+(0.0505−0.544i)T+(−0.982−0.183i)T2 1 + (0.0505 - 0.544i)T + (-0.982 - 0.183i)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

−9.846670454903261460976732275541, −9.275900294528494042917823540201, −8.752843074737745793661648803818, −7.36481273078247359207499605001, −6.61990805456316269122831702228, −5.85374769674114706976198542568, −4.60151714944977691422138597323, −3.29572481973458585869499104953, −2.49394383026765364546524919274, −1.42604584576712154741517710172, 1.52468738026870222163861167618, 2.31092646403877688716785443720, 4.48113470570963578340560204751, 5.19256625874237079512106997428, 5.97019417696875834406048012756, 6.59411310334288322142660291901, 7.72243896871272404695297503049, 8.565987097460109121066604046242, 9.194587685730624229071420237947, 9.892799718589182378220230690010

Graph of the ZZ-function along the critical line