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Properties

Label 2-1596-399.44-c0-0-0
Degree $2$
Conductor $1596$
Sign $-0.0482 - 0.998i$
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)3-s + (0.173 + 0.984i)7-s + (−0.939 + 0.342i)9-s + (1.76 − 0.642i)13-s + (−0.5 + 0.866i)19-s + (−0.939 + 0.342i)21-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s − 1.87·31-s + (−0.173 + 0.300i)37-s + (0.939 + 1.62i)39-s + (0.0603 + 0.342i)43-s + (−0.939 + 0.342i)49-s + (−0.939 − 0.342i)57-s + (0.266 + 0.223i)61-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (0.173 + 0.984i)7-s + (−0.939 + 0.342i)9-s + (1.76 − 0.642i)13-s + (−0.5 + 0.866i)19-s + (−0.939 + 0.342i)21-s + (0.766 + 0.642i)25-s + (−0.5 − 0.866i)27-s − 1.87·31-s + (−0.173 + 0.300i)37-s + (0.939 + 1.62i)39-s + (0.0603 + 0.342i)43-s + (−0.939 + 0.342i)49-s + (−0.939 − 0.342i)57-s + (0.266 + 0.223i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1596\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 19\)
Sign: $-0.0482 - 0.998i$
Analytic conductor: \(0.796507\)
Root analytic conductor: \(0.892472\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1596} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1596,\ (\ :0),\ -0.0482 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.192128809\)
\(L(\frac12)\) \(\approx\) \(1.192128809\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + 1.87T + T^{2} \)
37 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.711848779255420096206136497812, −8.884997559963138662183683786783, −8.545551400930209065350880723206, −7.67363321466695817183166611575, −6.23773209436227796620166596918, −5.70740360040077895755355283477, −4.91087616414156810167412372818, −3.74547319776564807523766518232, −3.15365994598183245386270133289, −1.80082444667734776313440011821, 1.00330758056010525985839703767, 2.07688843088773649105234915975, 3.41616083772618555694400357514, 4.19362631980387694711782121337, 5.43991246132106040028147934552, 6.46191603709004352906730829695, 6.91455107971598333496535764746, 7.74242428806683944381167847045, 8.648454629569131325985780672772, 9.041401235950326453023159179679

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