↑

Properties

Label 2-1260-1260.319-c0-0-0
Degree $2$
Conductor $1260$
Sign $0.400 - 0.916i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 0.999·12-s + 0.999·14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (−0.499 − 0.866i)20-s + 0.999·21-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + 0.999·12-s + 0.999·14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (−0.499 − 0.866i)20-s + 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7720104374\)
\(L(\frac12)\) \(\approx\) \(0.7720104374\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 - 2T + T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−9.934233195470402453857241428281, −9.219149705161012387304631671614, −8.758741674222259445643648626974, −7.33477807684882512473919782155, −6.70395650358065671509230206101, −5.92380666807666137504429657048, −5.12696519920336464650199977840, −4.36485917851616584587997433626, −3.05709666582319265364724981590, −1.09837548045772675850030893145, 1.19248004057010032279897048169, 2.37082301713530552093769824961, 2.98729812867865998246408079009, 4.72467365931598154474555656786, 5.59723659241164069648151583664, 6.45779692556955817692219893279, 7.27880299109105787386317056971, 8.335134978115309031605779819722, 9.079205349113882384530626256772, 9.676677298453753994436027749088

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