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Properties

Label 2-3800-152.43-c0-0-1
Degree $2$
Conductor $3800$
Sign $0.305 - 0.952i$
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  + (0.939 − 0.342i)2-s + (−0.266 + 1.50i)3-s + (0.766 − 0.642i)4-s + (0.266 + 1.50i)6-s + (0.500 − 0.866i)8-s + (−1.26 − 0.460i)9-s + (−0.766 + 1.32i)11-s + (0.766 + 1.32i)12-s + (0.173 − 0.984i)16-s + (1.87 − 0.684i)17-s − 1.34·18-s + (0.173 + 0.984i)19-s + (−0.266 + 1.50i)22-s + (1.17 + 0.984i)24-s + (0.266 − 0.460i)27-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.266 + 1.50i)3-s + (0.766 − 0.642i)4-s + (0.266 + 1.50i)6-s + (0.500 − 0.866i)8-s + (−1.26 − 0.460i)9-s + (−0.766 + 1.32i)11-s + (0.766 + 1.32i)12-s + (0.173 − 0.984i)16-s + (1.87 − 0.684i)17-s − 1.34·18-s + (0.173 + 0.984i)19-s + (−0.266 + 1.50i)22-s + (1.17 + 0.984i)24-s + (0.266 − 0.460i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.102447773\)
\(L(\frac12)\) \(\approx\) \(2.102447773\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 \)
19 \( 1 + (-0.173 - 0.984i)T \)
good3 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (-1.87 + 0.684i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.289165931608383071111764434265, −7.83786296659534332399894015220, −7.44589377671829500017087968194, −6.16804584422064722486882304729, −5.58004698720662338758040900864, −4.77454856969960233423241335878, −4.51786791022546050726692341993, −3.41579507485445526623998573046, −2.92995697128212972161041484612, −1.56978656524661406914173926497, 0.956308949984380190610881682794, 2.14661049649068149533747243579, 3.01734658281283502709430130126, 3.75925295809823066313639413377, 5.15200434434347178945456673992, 5.68384644973728031629369230449, 6.16837427989605134423301613210, 7.09415365738519213021655554667, 7.54076989017223937723669137483, 8.209955203676724815008982116220

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