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Properties

Label 2-1260-1260.499-c0-0-3
Degree $2$
Conductor $1260$
Sign $-0.281 + 0.959i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.499 + 0.866i)15-s + 16-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)20-s + (−0.499 + 0.866i)21-s + ⋯
L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.5 − 0.866i)14-s + (−0.499 + 0.866i)15-s + 16-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)20-s + (−0.499 + 0.866i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.281 + 0.959i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ -0.281 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.431508533\)
\(L(\frac12)\) \(\approx\) \(1.431508533\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
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 \( 1 + T + T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.885717195177434700378834997824, −8.459997025048691726897991249040, −7.84415019398192461326362710411, −6.93266978828341699837319671077, −6.38789688826467097175942736256, −5.36564384961272093052922983458, −4.55539876889089242112476614992, −3.71478996460746833015381137707, −2.37488084635141469393679416171, −0.981234201623604943288635837863, 2.36032236164369232968053724048, 3.37261966239823783958491165307, 3.95835651785243342472034659088, 5.08146992267197042385777617225, 5.91693142553783703794459450496, 6.44103096571371043453754541140, 7.45308424529244599162221830707, 8.434120176811553603136897318853, 9.759520217839065506552768014856, 10.09418742514605004559961610728

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