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Properties

Label 2-34e2-4.3-c0-0-1
Degree 22
Conductor 11561156
Sign 11
Analytic cond. 0.5769190.576919
Root an. cond. 0.7595510.759551
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 1.41·5-s − 8-s + 9-s − 1.41·10-s + 16-s − 18-s + 1.41·20-s + 1.00·25-s − 1.41·29-s − 32-s + 36-s − 1.41·37-s − 1.41·40-s − 1.41·41-s + 1.41·45-s + 49-s − 1.00·50-s + 1.41·58-s + 1.41·61-s + 64-s − 72-s − 1.41·73-s + 1.41·74-s + 1.41·80-s + 81-s + ⋯
L(s)  = 1  − 2-s + 4-s + 1.41·5-s − 8-s + 9-s − 1.41·10-s + 16-s − 18-s + 1.41·20-s + 1.00·25-s − 1.41·29-s − 32-s + 36-s − 1.41·37-s − 1.41·40-s − 1.41·41-s + 1.41·45-s + 49-s − 1.00·50-s + 1.41·58-s + 1.41·61-s + 64-s − 72-s − 1.41·73-s + 1.41·74-s + 1.41·80-s + 81-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 0.5769190.576919
Root analytic conductor: 0.7595510.759551
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1156(579,⋅)\chi_{1156} (579, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1156, ( :0), 1)(2,\ 1156,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.92244024660.9224402466
L(12)L(\frac12) ≈\approx 0.92244024660.9224402466
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+T 1 + T
17 1 1
good3 1−T2 1 - T^{2}
5 1−1.41T+T2 1 - 1.41T + T^{2}
7 1−T2 1 - T^{2}
11 1−T2 1 - T^{2}
13 1+T2 1 + T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1−T2 1 - T^{2}
37 1+1.41T+T2 1 + 1.41T + T^{2}
41 1+1.41T+T2 1 + 1.41T + T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−1.41T+T2 1 - 1.41T + T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1+1.41T+T2 1 + 1.41T + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+T2 1 + T^{2}
97 1−1.41T+T2 1 - 1.41T + 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

−10.07119591672476248371186517129, −9.253176209357900988401001777534, −8.616008782696301635635136454714, −7.44553303887263391257875235573, −6.83164465893778212385571895107, −5.95573941740973026735021173568, −5.14169447944175471979683650678, −3.61006690859387285815609488211, −2.23508563830816040785000016830, −1.49415910324576107748984374736, 1.49415910324576107748984374736, 2.23508563830816040785000016830, 3.61006690859387285815609488211, 5.14169447944175471979683650678, 5.95573941740973026735021173568, 6.83164465893778212385571895107, 7.44553303887263391257875235573, 8.616008782696301635635136454714, 9.253176209357900988401001777534, 10.07119591672476248371186517129

Graph of the ZZ-function along the critical line