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Properties

Label 2-3800-152.35-c0-0-2
Degree $2$
Conductor $3800$
Sign $0.756 - 0.654i$
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  + (−0.766 − 0.642i)2-s + (−1.76 + 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (0.500 − 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.766 + 1.32i)11-s + (−0.939 − 1.62i)12-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s − 2.53·18-s + (0.766 − 0.642i)19-s + (1.43 − 0.524i)22-s + (−0.326 + 1.85i)24-s + (−1.43 + 2.49i)27-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−1.76 + 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (0.500 − 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.766 + 1.32i)11-s + (−0.939 − 1.62i)12-s + (−0.939 + 0.342i)16-s + (0.766 + 0.642i)17-s − 2.53·18-s + (0.766 − 0.642i)19-s + (1.43 − 0.524i)22-s + (−0.326 + 1.85i)24-s + (−1.43 + 2.49i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.756 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4281996247\)
\(L(\frac12)\) \(\approx\) \(0.4281996247\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 \)
19 \( 1 + (-0.766 + 0.642i)T \)
good3 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
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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

−9.189671902870959749031167309800, −7.896707227416460516309998723906, −7.32679272453921567472527231223, −6.60599725274519839595105741581, −5.69689125861332785021017368765, −4.92410408957706829471385376034, −4.30105730120281373904677862784, −3.37659520587155956345329470465, −2.05552982932751691187882542096, −0.856843530204907589373333334017, 0.61065719934470144284918960642, 1.42759487613721642894819643118, 2.85705350107394372737111342598, 4.45452884919507975297115695901, 5.42904103617006082234934810520, 5.65097076719790309372732781847, 6.30588558214913846554464939925, 7.12175309694353353258385607611, 7.75634111008150022354005513035, 8.228451872926728703131366952024

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