↑

Properties

Label 2-2280-2280.1139-c0-0-8
Degree $2$
Conductor $2280$
Sign $i$
Analytic cond. $1.13786$
Root an. cond. $1.06670$
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  + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s − 5-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s − 2i·11-s + (−0.707 + 0.707i)12-s + 1.41i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 − 0.707i)18-s + 19-s + 1.00i·20-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s − 5-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s − 2i·11-s + (−0.707 + 0.707i)12-s + 1.41i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 − 0.707i)18-s + 19-s + 1.00i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $i$
Analytic conductor: \(1.13786\)
Root analytic conductor: \(1.06670\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3744014898\)
\(L(\frac12)\) \(\approx\) \(0.3744014898\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + T \)
19 \( 1 - T \)
good7 \( 1 + T^{2} \)
11 \( 1 + 2iT - T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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

−8.782793213847560318708865958240, −8.103830672841535332111044268784, −7.51191659289841465985318996560, −6.73691923181224540774244912393, −6.11914392004775590811596700980, −5.33544874940124108914983072740, −4.41666077373413560886090769362, −3.17841979441556657397835481272, −1.61658708890865965416116467196, −0.42151937943283536197836364234, 1.21103871898110647874194347277, 2.84037024928082009626347137519, 3.60863320013442869746374920478, 4.57632090422184291193173700152, 5.04228661423271317844112600621, 6.45948293277654186146233724407, 7.41551759960314149018763014763, 7.77815250033549944614597397674, 8.813402087965620523207356172014, 9.636586037390363913669083266580

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