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Properties

Label 2-975-975.731-c0-0-0
Degree 22
Conductor 975975
Sign 0.754+0.656i0.754 + 0.656i
Analytic cond. 0.4865880.486588
Root an. cond. 0.6975580.697558
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  + (−1.20 − 1.08i)2-s + (0.406 − 0.913i)3-s + (0.169 + 1.60i)4-s + (−0.951 + 0.309i)5-s + (−1.47 + 0.658i)6-s + (−0.5 + 0.866i)7-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (1.47 + 0.658i)10-s + (1.20 + 1.08i)11-s + (1.53 + 0.500i)12-s + (−0.309 + 0.951i)13-s + (1.53 − 0.499i)14-s + (−0.104 + 0.994i)15-s + (−0.406 − 0.913i)17-s + 1.61i·18-s + ⋯
L(s)  = 1  + (−1.20 − 1.08i)2-s + (0.406 − 0.913i)3-s + (0.169 + 1.60i)4-s + (−0.951 + 0.309i)5-s + (−1.47 + 0.658i)6-s + (−0.5 + 0.866i)7-s + (0.587 − 0.809i)8-s + (−0.669 − 0.743i)9-s + (1.47 + 0.658i)10-s + (1.20 + 1.08i)11-s + (1.53 + 0.500i)12-s + (−0.309 + 0.951i)13-s + (1.53 − 0.499i)14-s + (−0.104 + 0.994i)15-s + (−0.406 − 0.913i)17-s + 1.61i·18-s + ⋯

Functional equation

Λ(s)=(975s/2 ΓC(s) L(s)=((0.754+0.656i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(975s/2 ΓC(s) L(s)=((0.754+0.656i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 975975    =    3⋅52⋅133 \cdot 5^{2} \cdot 13
Sign: 0.754+0.656i0.754 + 0.656i
Analytic conductor: 0.4865880.486588
Root analytic conductor: 0.6975580.697558
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ975(731,⋅)\chi_{975} (731, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 975, ( :0), 0.754+0.656i)(2,\ 975,\ (\ :0),\ 0.754 + 0.656i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.48462733400.4846273340
L(12)L(\frac12) ≈\approx 0.48462733400.4846273340
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)
bad3 1+(−0.406+0.913i)T 1 + (-0.406 + 0.913i)T
5 1+(0.951−0.309i)T 1 + (0.951 - 0.309i)T
13 1+(0.309−0.951i)T 1 + (0.309 - 0.951i)T
good2 1+(1.20+1.08i)T+(0.104+0.994i)T2 1 + (1.20 + 1.08i)T + (0.104 + 0.994i)T^{2}
7 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
11 1+(−1.20−1.08i)T+(0.104+0.994i)T2 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2}
17 1+(0.406+0.913i)T+(−0.669+0.743i)T2 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2}
19 1+(−0.913+0.406i)T+(0.669−0.743i)T2 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2}
23 1+(−0.743−0.669i)T+(0.104+0.994i)T2 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2}
29 1+(−0.669−0.743i)T2 1 + (-0.669 - 0.743i)T^{2}
31 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
37 1+(−0.978+0.207i)T+(0.913−0.406i)T2 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2}
41 1+(−0.207−0.978i)T+(−0.913+0.406i)T2 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2}
43 1+(0.309−0.535i)T+(−0.5−0.866i)T2 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2}
47 1+(−0.363−0.5i)T+(−0.309+0.951i)T2 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2}
53 1+(−0.951−1.30i)T+(−0.309+0.951i)T2 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2}
59 1+(0.743−0.669i)T+(0.104−0.994i)T2 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2}
61 1+(0.978+0.207i)T+(0.913+0.406i)T2 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2}
67 1+(0.0646−0.614i)T+(−0.978−0.207i)T2 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2}
71 1+(0.978−0.207i)T2 1 + (0.978 - 0.207i)T^{2}
73 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
79 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
83 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
89 1+(0.743+0.669i)T+(0.104+0.994i)T2 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2}
97 1+(−0.0646−0.614i)T+(−0.978+0.207i)T2 1 + (-0.0646 - 0.614i)T + (-0.978 + 0.207i)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

−9.788834406496960241744628080176, −9.094449903310391492049129228089, −8.968899856717625176374250238828, −7.52917578029189169580675006564, −7.28691177758162888329487501445, −6.24717886957619948038785543022, −4.46891056564961789220597746543, −3.17515446073533095833048346122, −2.51370663415033563705692659390, −1.29358138860036969335712234091, 0.76847739084472296387364214975, 3.36003344122042134619202223480, 3.95243065873127905691047727228, 5.26402756143772350549754000219, 6.27705794567346312785233075984, 7.20977700351227510639472187847, 7.993319252070138997934276544916, 8.626219164981535591849168763781, 9.181921421859371707868764622201, 10.11742608075968704116609294289

Graph of the ZZ-function along the critical line