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Properties

Label 2-3800-152.43-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.434531710\)
\(L(\frac12)\) \(\approx\) \(1.434531710\)
\(L(1)\) 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

−8.829905757269589809860931367512, −8.115010700093850905516033495186, −7.18301714958166691180698294125, −6.50028659185518790706407975402, −5.97872627227159333589546515521, −5.40532529897249968540355285485, −4.24489676176628078381684704960, −3.45233309881807747028732558078, −2.95830261580809088521850191848, −1.06351610639851759491848549362, 1.02515826655293376180756107969, 1.88485608154986027429513077359, 3.19996574208445011845688936744, 3.92595724022264818436026530327, 4.25987924244677246631321984233, 5.35522469890056691009817623294, 6.47463303416434809061154326665, 6.80483958023093188544392454453, 7.64131112177382238630409412537, 8.984131274526831981284482561799

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