Properties

Label 2-975-975.731-c0-0-0
Degree $2$
Conductor $975$
Sign $0.754 + 0.656i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.754 + 0.656i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.754 + 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4846273340\)
\(L(\frac12)\) \(\approx\) \(0.4846273340\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.406 + 0.913i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (1.20 + 1.08i)T + (0.104 + 0.994i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \)
17 \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \)
19 \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \)
23 \( 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2} \)
29 \( 1 + (-0.669 - 0.743i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
41 \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \)
43 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.743 - 0.669i)T + (0.104 - 0.994i)T^{2} \)
61 \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \)
67 \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \)
71 \( 1 + (0.978 - 0.207i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \)
97 \( 1 + (-0.0646 - 0.614i)T + (-0.978 + 0.207i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line