Properties

Label 2-3800-152.43-c0-0-4
Degree $2$
Conductor $3800$
Sign $-0.135 + 0.990i$
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  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.32 − 0.766i)7-s + (0.866 + 0.500i)8-s + (0.939 + 0.342i)9-s + (0.939 − 1.62i)11-s + (−0.984 + 0.173i)13-s + (−1.17 − 0.984i)14-s + (0.173 − 0.984i)16-s i·18-s + (0.939 − 0.342i)19-s + (−1.85 − 0.326i)22-s + (−0.223 − 0.266i)23-s + (0.5 + 0.866i)26-s + (−0.524 + 1.43i)28-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.32 − 0.766i)7-s + (0.866 + 0.500i)8-s + (0.939 + 0.342i)9-s + (0.939 − 1.62i)11-s + (−0.984 + 0.173i)13-s + (−1.17 − 0.984i)14-s + (0.173 − 0.984i)16-s i·18-s + (0.939 − 0.342i)19-s + (−1.85 − 0.326i)22-s + (−0.223 − 0.266i)23-s + (0.5 + 0.866i)26-s + (−0.524 + 1.43i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.135 + 0.990i$
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: \((2,\ 3800,\ (\ :0),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.316358888\)
\(L(\frac12)\) \(\approx\) \(1.316358888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-0.939 + 0.342i)T \)
good3 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.223 + 0.266i)T + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 0.347iT - T^{2} \)
41 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.984 + 1.17i)T + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.766 - 0.642i)T^{2} \)
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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

−8.459424019596921319025735996074, −7.87551381704565072707940793792, −7.34185625306633107385137117118, −6.34433816117544612258188447220, −5.01078077269123625308352239872, −4.63804786731843626211659055779, −3.78629895476868934294499498766, −2.91951261642478425809591809181, −1.65871990825826427361377317461, −1.02978445708095806744248117483, 1.44429319361481778165190947211, 2.04905647114621949797222304320, 3.81961640945775764132732532941, 4.65504221496309316328198481292, 5.05202578907980099472286711061, 5.92465822453296900485343627978, 6.97518770134183356462937722738, 7.36049942651516457237368010889, 7.932739823396552466772707135940, 8.899817632867806001294316819252

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