↑

Properties

Label 2-1596-399.158-c0-0-0
Degree $2$
Conductor $1596$
Sign $0.580 + 0.814i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (1.76 − 0.642i)13-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)21-s + (0.173 − 0.984i)25-s + (−0.500 − 0.866i)27-s + (0.939 + 1.62i)31-s + (−0.173 − 0.300i)37-s + (0.939 − 1.62i)39-s + (0.0603 + 0.342i)43-s + (0.766 − 0.642i)49-s + (−0.939 − 0.342i)57-s + (−0.326 + 0.118i)61-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (1.76 − 0.642i)13-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)21-s + (0.173 − 0.984i)25-s + (−0.500 − 0.866i)27-s + (0.939 + 1.62i)31-s + (−0.173 − 0.300i)37-s + (0.939 − 1.62i)39-s + (0.0603 + 0.342i)43-s + (0.766 − 0.642i)49-s + (−0.939 − 0.342i)57-s + (−0.326 + 0.118i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1596\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(0.796507\)
Root analytic conductor: \(0.892472\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1596} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1596,\ (\ :0),\ 0.580 + 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.371336220\)
\(L(\frac12)\) \(\approx\) \(1.371336220\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.247324668212553743743617671852, −8.596257836783332825803158150281, −8.157895194682684265717833580212, −6.89473212983980318632633558329, −6.47289263817039806272380411393, −5.63657190610613069031064021000, −4.21235357249640903587294318871, −3.26704066021766623011223218785, −2.59421164751314955964943914624, −1.12815082198037714674626673625, 1.66063264088765336808306138335, 3.01887537152401655848876773840, 3.78418982636600759427768294568, 4.38462894984632236837996977129, 5.77113364906848445468082126002, 6.42211683303129298430152562953, 7.45923732720615871418503545453, 8.304799572345683847647319480691, 8.997982603102878012931364259112, 9.625494326450565437900863962339

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