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Properties

Label 2-760-760.379-c1-0-37
Degree $2$
Conductor $760$
Sign $0.922 - 0.384i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 − 0.687i)2-s + 2.17·3-s + (1.05 + 1.70i)4-s + (−2.19 − 0.451i)5-s + (−2.68 − 1.49i)6-s − 2.03·7-s + (−0.131 − 2.82i)8-s + 1.71·9-s + (2.39 + 2.06i)10-s + 3.06·11-s + (2.28 + 3.69i)12-s + 6.48i·13-s + (2.51 + 1.39i)14-s + (−4.75 − 0.980i)15-s + (−1.78 + 3.58i)16-s − 0.290i·17-s + ⋯
L(s)  = 1  + (−0.873 − 0.486i)2-s + 1.25·3-s + (0.526 + 0.850i)4-s + (−0.979 − 0.201i)5-s + (−1.09 − 0.610i)6-s − 0.768·7-s + (−0.0465 − 0.998i)8-s + 0.572·9-s + (0.757 + 0.652i)10-s + 0.923·11-s + (0.660 + 1.06i)12-s + 1.79i·13-s + (0.671 + 0.373i)14-s + (−1.22 − 0.253i)15-s + (−0.445 + 0.895i)16-s − 0.0704i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.922 - 0.384i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.922 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17759 + 0.235766i\)
\(L(\frac12)\) \(\approx\) \(1.17759 + 0.235766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 + 0.687i)T \)
5 \( 1 + (2.19 + 0.451i)T \)
19 \( 1 + (-2.28 - 3.71i)T \)
good3 \( 1 - 2.17T + 3T^{2} \)
7 \( 1 + 2.03T + 7T^{2} \)
11 \( 1 - 3.06T + 11T^{2} \)
13 \( 1 - 6.48iT - 13T^{2} \)
17 \( 1 + 0.290iT - 17T^{2} \)
23 \( 1 - 7.22T + 23T^{2} \)
29 \( 1 - 7.49T + 29T^{2} \)
31 \( 1 - 1.72T + 31T^{2} \)
37 \( 1 - 0.791iT - 37T^{2} \)
41 \( 1 - 8.58iT - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 - 0.0753T + 47T^{2} \)
53 \( 1 - 3.15iT - 53T^{2} \)
59 \( 1 - 6.29iT - 59T^{2} \)
61 \( 1 + 6.53iT - 61T^{2} \)
67 \( 1 + 0.557T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 - 7.28T + 79T^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 + 0.581iT - 89T^{2} \)
97 \( 1 - 3.90T + 97T^{2} \)
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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

−10.08110702517243200199983855056, −9.231951171599993857508822122596, −8.893078580888729320304296457890, −8.114261298302553123114570244816, −7.13412606558659190915356728719, −6.57327748074428088439355610322, −4.38000588024525109293482374851, −3.59123112930199706061160921845, −2.81498226641584950353086630479, −1.36479410539796686978738169482, 0.791778602721877856511806434919, 2.83862677462363350339929472118, 3.30317416009019531479301406563, 4.86757336622583011913586005726, 6.24928740425844464582960103076, 7.16003826443944956047495582791, 7.80289940163978540479846571118, 8.614608280111780782985447538673, 9.131035642261316373754677189782, 10.03879961369901856898998459442

Graph of the $Z$-function along the critical line