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Properties

Label 2-1596-1596.1259-c0-0-0
Degree $2$
Conductor $1596$
Sign $0.612 - 0.790i$
Analytic cond. $0.796507$
Root an. cond. $0.892472$
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  + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.266 − 0.223i)5-s + (−0.173 − 0.984i)6-s + (0.5 + 0.866i)7-s + (−0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.326 + 0.118i)10-s + (0.939 − 1.62i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)14-s + (0.266 − 0.223i)15-s + (0.173 − 0.984i)16-s + (1.43 − 0.524i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.266 − 0.223i)5-s + (−0.173 − 0.984i)6-s + (0.5 + 0.866i)7-s + (−0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.326 + 0.118i)10-s + (0.939 − 1.62i)11-s + (0.5 + 0.866i)12-s + (−0.766 − 0.642i)14-s + (0.266 − 0.223i)15-s + (0.173 − 0.984i)16-s + (1.43 − 0.524i)17-s + 0.999·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1596\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(0.796507\)
Root analytic conductor: \(0.892472\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1596} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1596,\ (\ :0),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7456253153\)
\(L(\frac12)\) \(\approx\) \(0.7456253153\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
good5 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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

−9.495654226234039403376332396294, −9.097258242607008585254979642532, −8.115413052975519556285253904875, −7.84157046251323533070334156465, −6.14856531751945901519292050489, −5.86537148172826594810429442065, −5.01703874558420879152706773831, −3.69800634562293872714428284296, −2.79077923316672954495189486746, −1.08399056496362065400811103529, 1.17617746979941180312282470185, 1.92029342847969771298959186847, 3.27418107742079849620701050926, 4.26157081468873519247673860345, 5.61882782181612992657581613530, 6.76048303480544966398923942305, 7.32422447326384033356511582907, 7.67533334357434650648319442670, 8.577219646658225390249599104151, 9.591066306867864577701498041535

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