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Properties

Label 2-1564-1564.1223-c0-0-1
Degree 22
Conductor 15641564
Sign −0.952−0.305i-0.952 - 0.305i
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 + (0.273 − 0.0801i)3-s + (−0.654 + 0.755i)4-s + (0.186 + 0.215i)6-s + (0.186 + 1.29i)7-s + (−0.959 − 0.281i)8-s + (−0.773 + 0.496i)9-s + (−0.797 + 1.74i)11-s + (−0.118 + 0.258i)12-s + (0.186 − 1.29i)13-s + (−1.10 + 0.708i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (−0.773 − 0.496i)18-s + (0.154 + 0.339i)21-s − 1.91·22-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (0.273 − 0.0801i)3-s + (−0.654 + 0.755i)4-s + (0.186 + 0.215i)6-s + (0.186 + 1.29i)7-s + (−0.959 − 0.281i)8-s + (−0.773 + 0.496i)9-s + (−0.797 + 1.74i)11-s + (−0.118 + 0.258i)12-s + (0.186 − 1.29i)13-s + (−1.10 + 0.708i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)17-s + (−0.773 − 0.496i)18-s + (0.154 + 0.339i)21-s − 1.91·22-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 15641564    =    22⋅17⋅232^{2} \cdot 17 \cdot 23
Sign: −0.952−0.305i-0.952 - 0.305i
Analytic conductor: 0.7805370.780537
Root analytic conductor: 0.8834800.883480
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1564(1223,⋅)\chi_{1564} (1223, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1564, ( :0), −0.952−0.305i)(2,\ 1564,\ (\ :0),\ -0.952 - 0.305i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0873047571.087304757
L(12)L(\frac12) ≈\approx 1.0873047571.087304757
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+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
good3 1+(−0.273+0.0801i)T+(0.841−0.540i)T2 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2}
5 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
7 1+(−0.186−1.29i)T+(−0.959+0.281i)T2 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2}
11 1+(0.797−1.74i)T+(−0.654−0.755i)T2 1 + (0.797 - 1.74i)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+(−1.84−0.540i)T+(0.841+0.540i)T2 1 + (-1.84 - 0.540i)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+(−0.830−1.81i)T+(−0.654+0.755i)T2 1 + (-0.830 - 1.81i)T + (-0.654 + 0.755i)T^{2}
73 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
79 1+(0.118−0.822i)T+(−0.959−0.281i)T2 1 + (0.118 - 0.822i)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

−9.747900285421762962044915203636, −8.897829516309000022327402744948, −8.255282862975311927596193657359, −7.68292575983350579818578258237, −6.79593376406128167915606683837, −5.74137870428161482626032827821, −5.17065668553531068607147749645, −4.50840165735720587862679759654, −2.87576759909559420010341410952, −2.45209876422008420674718989062, 0.71051724044555093964614809119, 2.21016625017147395214686801366, 3.34702221310977595260068398682, 3.94320298958285459985715363231, 4.82929198538778766359186710720, 6.05103647327584917322833506204, 6.50079884127105418492892363978, 8.053782763593601696968171126958, 8.506485343959601457288436794966, 9.369456989991438851675285674406

Graph of the ZZ-function along the critical line