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Properties

Label 2-1564-1564.1291-c0-0-0
Degree 22
Conductor 15641564
Sign −0.994+0.105i-0.994 + 0.105i
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  + (0.142 + 0.989i)2-s + (−0.449 + 0.983i)3-s + (−0.959 + 0.281i)4-s + (−1.03 − 0.304i)6-s + (0.474 − 0.304i)7-s + (−0.415 − 0.909i)8-s + (−0.110 − 0.127i)9-s + (−0.258 + 1.80i)11-s + (0.153 − 1.07i)12-s + (1.61 + 1.03i)13-s + (0.368 + 0.425i)14-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + (0.110 − 0.127i)18-s + (0.0867 + 0.603i)21-s − 1.81·22-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)2-s + (−0.449 + 0.983i)3-s + (−0.959 + 0.281i)4-s + (−1.03 − 0.304i)6-s + (0.474 − 0.304i)7-s + (−0.415 − 0.909i)8-s + (−0.110 − 0.127i)9-s + (−0.258 + 1.80i)11-s + (0.153 − 1.07i)12-s + (1.61 + 1.03i)13-s + (0.368 + 0.425i)14-s + (0.841 − 0.540i)16-s + (−0.959 − 0.281i)17-s + (0.110 − 0.127i)18-s + (0.0867 + 0.603i)21-s − 1.81·22-s + ⋯

Functional equation

Λ(s)=(1564s/2 ΓC(s) L(s)=((−0.994+0.105i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1564s/2 ΓC(s) L(s)=((−0.994+0.105i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15641564    =    22⋅17⋅232^{2} \cdot 17 \cdot 23
Sign: −0.994+0.105i-0.994 + 0.105i
Analytic conductor: 0.7805370.780537
Root analytic conductor: 0.8834800.883480
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1564(1291,⋅)\chi_{1564} (1291, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1564, ( :0), −0.994+0.105i)(2,\ 1564,\ (\ :0),\ -0.994 + 0.105i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91397326730.9139732673
L(12)L(\frac12) ≈\approx 0.91397326730.9139732673
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+(−0.142−0.989i)T 1 + (-0.142 - 0.989i)T
17 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
23 1+(0.281−0.959i)T 1 + (0.281 - 0.959i)T
good3 1+(0.449−0.983i)T+(−0.654−0.755i)T2 1 + (0.449 - 0.983i)T + (-0.654 - 0.755i)T^{2}
5 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
7 1+(−0.474+0.304i)T+(0.415−0.909i)T2 1 + (-0.474 + 0.304i)T + (0.415 - 0.909i)T^{2}
11 1+(0.258−1.80i)T+(−0.959−0.281i)T2 1 + (0.258 - 1.80i)T + (-0.959 - 0.281i)T^{2}
13 1+(−1.61−1.03i)T+(0.415+0.909i)T2 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2}
19 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
29 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
31 1+(0.755+1.65i)T+(−0.654+0.755i)T2 1 + (0.755 + 1.65i)T + (-0.654 + 0.755i)T^{2}
37 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
41 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)T^{2}
43 1+(0.654+0.755i)T2 1 + (0.654 + 0.755i)T^{2}
47 1−T2 1 - T^{2}
53 1+(−0.698+0.449i)T+(0.415−0.909i)T2 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2}
59 1+(−0.415−0.909i)T2 1 + (-0.415 - 0.909i)T^{2}
61 1+(0.654−0.755i)T2 1 + (0.654 - 0.755i)T^{2}
67 1+(0.959−0.281i)T2 1 + (0.959 - 0.281i)T^{2}
71 1+(−0.959+0.281i)T2 1 + (-0.959 + 0.281i)T^{2}
73 1+(−0.841+0.540i)T2 1 + (-0.841 + 0.540i)T^{2}
79 1+(−1.66−1.07i)T+(0.415+0.909i)T2 1 + (-1.66 - 1.07i)T + (0.415 + 0.909i)T^{2}
83 1+(0.142−0.989i)T2 1 + (0.142 - 0.989i)T^{2}
89 1+(0.345−0.755i)T+(−0.654−0.755i)T2 1 + (0.345 - 0.755i)T + (-0.654 - 0.755i)T^{2}
97 1+(0.142+0.989i)T2 1 + (0.142 + 0.989i)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

−9.681438273800762845590911139544, −9.441919269180673483420851403699, −8.380061940050689661121015314301, −7.54105606170735192686114547701, −6.80509154920857033278653378198, −5.92468673080924596955677204480, −5.00567422516814376195407313838, −4.21668349775703686089923095734, −3.99634996695436506228547079257, −1.92130090842809939313893319472, 0.78757257141016385497689176826, 1.76158311369799761846198673027, 3.09029979995766359027079529528, 3.82502213310086017982250102024, 5.23692234994416052834669995883, 5.86291802867704184225898914119, 6.53032717982026739722228844521, 7.893764595650719499657709490879, 8.576267652827193586269019311343, 8.961181071718193777426819751860

Graph of the ZZ-function along the critical line