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Properties

Label 2-3800-152.131-c0-0-4
Degree $2$
Conductor $3800$
Sign $-0.755 - 0.654i$
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  + (0.173 − 0.984i)2-s + (−1.43 − 1.20i)3-s + (−0.939 − 0.342i)4-s + (−1.43 + 1.20i)6-s + (−0.5 + 0.866i)8-s + (0.439 + 2.49i)9-s + (0.939 − 1.62i)11-s + (0.939 + 1.62i)12-s + (0.766 + 0.642i)16-s + (0.347 − 1.96i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (−1.43 − 1.20i)22-s + (1.76 − 0.642i)24-s + (1.43 − 2.49i)27-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−1.43 − 1.20i)3-s + (−0.939 − 0.342i)4-s + (−1.43 + 1.20i)6-s + (−0.5 + 0.866i)8-s + (0.439 + 2.49i)9-s + (0.939 − 1.62i)11-s + (0.939 + 1.62i)12-s + (0.766 + 0.642i)16-s + (0.347 − 1.96i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (−1.43 − 1.20i)22-s + (1.76 − 0.642i)24-s + (1.43 − 2.49i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7453743011\)
\(L(\frac12)\) \(\approx\) \(0.7453743011\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 \)
19 \( 1 + (-0.766 + 0.642i)T \)
good3 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
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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

−8.182229085388036435641949091506, −7.37443338735599821996370545621, −6.61158215836580888091489957682, −5.85063654151586953630955142387, −5.31264056752075010775153220348, −4.59464964680142459304021753520, −3.33918082793443330842727523379, −2.48263908152278517162202162388, −1.18107492302437985806795279995, −0.63099925734261402372765677228, 1.38416345661263996789630470374, 3.54264714475081828678712620775, 4.15113163544362188062074124276, 4.59839509085462603063832404023, 5.56643260535043660021912101925, 5.96843985615583280750393607032, 6.72602439125091755802305750393, 7.37484360300261325205834763985, 8.419513314305985011782717282935, 9.267958402196201706520548699343

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