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Properties

Label 2-3800-152.131-c0-0-1
Degree $2$
Conductor $3800$
Sign $0.0158 - 0.999i$
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  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−1.62 + 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.173 + 0.300i)11-s + (0.642 + 0.766i)13-s + (−1.76 + 0.642i)14-s + (0.766 + 0.642i)16-s − 0.999i·18-s + (−0.173 + 0.984i)19-s + (−0.223 + 0.266i)22-s + (−0.524 + 1.43i)23-s + (0.5 + 0.866i)26-s + (−1.85 + 0.326i)28-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−1.62 + 0.939i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.173 + 0.300i)11-s + (0.642 + 0.766i)13-s + (−1.76 + 0.642i)14-s + (0.766 + 0.642i)16-s − 0.999i·18-s + (−0.173 + 0.984i)19-s + (−0.223 + 0.266i)22-s + (−0.524 + 1.43i)23-s + (0.5 + 0.866i)26-s + (−1.85 + 0.326i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.0158 - 0.999i$
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: \((2,\ 3800,\ (\ :0),\ 0.0158 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896470727\)
\(L(\frac12)\) \(\approx\) \(1.896470727\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 - 0.173i)T \)
5 \( 1 \)
19 \( 1 + (0.173 - 0.984i)T \)
good3 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.53iT - T^{2} \)
41 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.642 + 1.76i)T + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.939 - 0.342i)T^{2} \)
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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

−8.835804266530214422716900350144, −8.092098597735882409760399676329, −6.96546758081173651114206750943, −6.50524459964501095070624854249, −5.91800571150585852249597302897, −5.34841066461703795745045914192, −4.00480182367781979592688722360, −3.55598764224215274574476419270, −2.80366746769176514471843634747, −1.71637813427509189513980870520, 0.74751150436094506008877153765, 2.44514766464527012564157558600, 2.97341384824334508680579613313, 3.94143699379803011195751617190, 4.48901229309227030261335219642, 5.67897811625941526547607156317, 6.04239541180613775939514651349, 6.99351047441964366713600417438, 7.41910970068768186657221465644, 8.406013416656313386893228889050

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