Properties

Label 2-3800-760.139-c0-0-0
Degree $2$
Conductor $3800$
Sign $-0.607 + 0.794i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
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  + (0.642 + 0.766i)2-s + (−0.524 + 1.43i)3-s + (−0.173 + 0.984i)4-s + (−1.43 + 0.524i)6-s + (−0.866 + 0.500i)8-s + (−1.03 − 0.866i)9-s + (0.939 + 1.62i)11-s + (−1.32 − 0.766i)12-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s − 1.34i·18-s + (−0.173 + 0.984i)19-s + (−0.642 + 1.76i)22-s + (−0.266 − 1.50i)24-s + (0.460 − 0.266i)27-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.524 + 1.43i)3-s + (−0.173 + 0.984i)4-s + (−1.43 + 0.524i)6-s + (−0.866 + 0.500i)8-s + (−1.03 − 0.866i)9-s + (0.939 + 1.62i)11-s + (−1.32 − 0.766i)12-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s − 1.34i·18-s + (−0.173 + 0.984i)19-s + (−0.642 + 1.76i)22-s + (−0.266 − 1.50i)24-s + (0.460 − 0.266i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.607 + 0.794i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.607 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229087059\)
\(L(\frac12)\) \(\approx\) \(1.229087059\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 - 0.766i)T \)
5 \( 1 \)
19 \( 1 + (0.173 - 0.984i)T \)
good3 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.642 + 1.76i)T + (-0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.223 - 0.266i)T + (-0.173 + 0.984i)T^{2} \)
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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

−9.296195182517455550458953697917, −8.459730857418184125065479628745, −7.48442354560821613501645261873, −6.78042704776828942516978160864, −6.10939846765994611163540489138, −5.16906595142857604713880411085, −4.68260760100712480164423913007, −4.07943053464766651834825326893, −3.41671489553824395507744801762, −2.09458520194983912617857706974, 0.60888182866784265040424091979, 1.51050298676087786511091072176, 2.43548109833514510922437320191, 3.41882111367050570248803044137, 4.26858406646877009897720155347, 5.40952627679982851534070383057, 5.96769643401102341492227937276, 6.62906988442498469553413014394, 7.06623866303850991088800936899, 8.474686016589772901744160567884

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