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Properties

Label 2-3800-152.139-c0-0-6
Degree $2$
Conductor $3800$
Sign $-0.944 + 0.327i$
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 + (−0.326 − 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (−0.500 − 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.173 − 0.300i)11-s + (−0.173 + 0.300i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s − 0.879·18-s + (−0.939 + 0.342i)19-s + (−0.326 − 0.118i)22-s + (0.0603 + 0.342i)24-s + (0.326 + 0.565i)27-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.326 − 0.118i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)6-s + (−0.500 − 0.866i)8-s + (−0.673 − 0.565i)9-s + (−0.173 − 0.300i)11-s + (−0.173 + 0.300i)12-s + (−0.939 − 0.342i)16-s + (1.53 − 1.28i)17-s − 0.879·18-s + (−0.939 + 0.342i)19-s + (−0.326 − 0.118i)22-s + (0.0603 + 0.342i)24-s + (0.326 + 0.565i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.944 + 0.327i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.334475423\)
\(L(\frac12)\) \(\approx\) \(1.334475423\)
\(L(1)\) 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 \( 1 + (0.939 - 0.342i)T \)
good3 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
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 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
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 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
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

−8.504005184735619740129174072947, −7.42199575729730029518229857722, −6.68209315383449728255707631904, −5.85309521639287561963078924282, −5.40358468396883439418968532947, −4.57902630061509221961705527343, −3.47054503241654298779352632801, −3.03076114885464023737334460883, −1.85470866296079420774986080170, −0.58391819050418152457507952929, 1.84220947526851715491844553185, 2.93756536223408448999164366605, 3.70621111949113716651508636461, 4.71206751077087603506419840899, 5.20691098118497750988398324286, 6.11470862591100530793225752342, 6.45960219681886280507363385454, 7.63110860840224826233933024989, 8.069387725411025898371851100098, 8.668514962755325747850032143370

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