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Properties

Label 2-1564-1564.1087-c0-0-2
Degree 22
Conductor 15641564
Sign −0.305−0.952i-0.305 - 0.952i
Analytic cond. 0.7805370.780537
Root an. cond. 0.8834800.883480
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  + (−0.415 + 0.909i)2-s + (1.89 + 0.557i)3-s + (−0.654 − 0.755i)4-s + (−1.29 + 1.49i)6-s + (−0.215 + 1.49i)7-s + (0.959 − 0.281i)8-s + (2.45 + 1.57i)9-s + (−0.234 − 0.512i)11-s + (−0.822 − 1.80i)12-s + (−0.186 − 1.29i)13-s + (−1.27 − 0.817i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (−2.45 + 1.57i)18-s + (−1.24 + 2.72i)21-s + 0.563·22-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (1.89 + 0.557i)3-s + (−0.654 − 0.755i)4-s + (−1.29 + 1.49i)6-s + (−0.215 + 1.49i)7-s + (0.959 − 0.281i)8-s + (2.45 + 1.57i)9-s + (−0.234 − 0.512i)11-s + (−0.822 − 1.80i)12-s + (−0.186 − 1.29i)13-s + (−1.27 − 0.817i)14-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + (−2.45 + 1.57i)18-s + (−1.24 + 2.72i)21-s + 0.563·22-s + ⋯

Functional equation

Λ(s)=(1564s/2 ΓC(s) L(s)=((−0.305−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1564s/2 ΓC(s) L(s)=((−0.305−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15641564    =    22⋅17⋅232^{2} \cdot 17 \cdot 23
Sign: −0.305−0.952i-0.305 - 0.952i
Analytic conductor: 0.7805370.780537
Root analytic conductor: 0.8834800.883480
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1564(1087,⋅)\chi_{1564} (1087, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1564, ( :0), −0.305−0.952i)(2,\ 1564,\ (\ :0),\ -0.305 - 0.952i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5601619401.560161940
L(12)L(\frac12) ≈\approx 1.5601619401.560161940
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+(0.415−0.909i)T 1 + (0.415 - 0.909i)T
17 1+(0.654−0.755i)T 1 + (0.654 - 0.755i)T
23 1+(0.755+0.654i)T 1 + (0.755 + 0.654i)T
good3 1+(−1.89−0.557i)T+(0.841+0.540i)T2 1 + (-1.89 - 0.557i)T + (0.841 + 0.540i)T^{2}
5 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
7 1+(0.215−1.49i)T+(−0.959−0.281i)T2 1 + (0.215 - 1.49i)T + (-0.959 - 0.281i)T^{2}
11 1+(0.234+0.512i)T+(−0.654+0.755i)T2 1 + (0.234 + 0.512i)T + (-0.654 + 0.755i)T^{2}
13 1+(0.186+1.29i)T+(−0.959+0.281i)T2 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2}
19 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
29 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
31 1+(0.540−0.158i)T+(0.841−0.540i)T2 1 + (0.540 - 0.158i)T + (0.841 - 0.540i)T^{2}
37 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
41 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
43 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
47 1−T2 1 - T^{2}
53 1+(−0.273+1.89i)T+(−0.959−0.281i)T2 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2}
59 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
61 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
67 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
71 1+(−0.654−0.755i)T2 1 + (-0.654 - 0.755i)T^{2}
73 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
79 1+(−0.258−1.80i)T+(−0.959+0.281i)T2 1 + (-0.258 - 1.80i)T + (-0.959 + 0.281i)T^{2}
83 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
89 1+(1.84+0.540i)T+(0.841+0.540i)T2 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2}
97 1+(−0.415+0.909i)T2 1 + (-0.415 + 0.909i)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

−9.632107505430983542454933572559, −8.624679519049711754216977210165, −8.510502013640778459804720491821, −7.938142060747297639536298179298, −6.83578934292064584682563097815, −5.80197678482934667547505127992, −4.95875537786533063104343341717, −3.91731601232397304558421219610, −2.86051475055511532780415747823, −2.04608667435664269152036601236, 1.35303133368798051177019477375, 2.18505487401638818558808968873, 3.19197036492448064429258159371, 4.02189972163525548459817766169, 4.51909018057337448954025778859, 6.82129951109202634947175070113, 7.33595376715233201689207732836, 7.76266478507750659527887788185, 8.848061322639479548719207053368, 9.356225275580860291842039675653

Graph of the ZZ-function along the critical line