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Properties

Label 2-1596-1596.587-c0-0-0
Degree $2$
Conductor $1596$
Sign $-0.513 + 0.858i$
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  + (−0.173 + 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−1.43 + 0.524i)5-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.266 − 1.50i)10-s + (0.173 − 0.300i)11-s + (0.499 + 0.866i)12-s + (−0.939 + 0.342i)14-s + (1.43 + 0.524i)15-s + (0.766 + 0.642i)16-s + (−0.326 + 1.85i)17-s − 18-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−1.43 + 0.524i)5-s + (0.766 − 0.642i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (−0.266 − 1.50i)10-s + (0.173 − 0.300i)11-s + (0.499 + 0.866i)12-s + (−0.939 + 0.342i)14-s + (1.43 + 0.524i)15-s + (0.766 + 0.642i)16-s + (−0.326 + 1.85i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1596\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.513 + 0.858i$
Analytic conductor: \(0.796507\)
Root analytic conductor: \(0.892472\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1596} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1596,\ (\ :0),\ -0.513 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05256434447\)
\(L(\frac12)\) \(\approx\) \(0.05256434447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
good5 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
11 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.87T + T^{2} \)
41 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
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

−10.32022389599790136420280832115, −8.773431973775991045648446784726, −8.398557899887585175419486514911, −7.67947634432716005779566158573, −6.93565487674500504091367410310, −6.23892351543541037180721008959, −5.46416866065203337396913348845, −4.49691067266903126891791495420, −3.65094745421155846495329350254, −1.89383736951225534150128550728, 0.05202601799244217133396259807, 1.42176196092258843534915476616, 3.36999891857376333387808145704, 3.94260904501466299474705932569, 4.73936922071496572740076030852, 5.19593962246463851285593950221, 6.86635428764257824137643178099, 7.64400306061263336795372501863, 8.415181829659165421918642781708, 9.245597442456064207946133161033

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