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Properties

Label 2-1260-1260.1159-c0-0-2
Degree $2$
Conductor $1260$
Sign $0.971 + 0.235i$
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 + 7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + 14-s + 0.999·15-s + 16-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)20-s + (−0.5 − 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 + 7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + 14-s + 0.999·15-s + 16-s + (−0.499 + 0.866i)18-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.725884569\)
\(L(\frac12)\) \(\approx\) \(1.725884569\)
\(L(1)\) 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 \( 1 - T \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + T + T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.56333807576407540861189283847, −8.734327343659711394405046560764, −7.83574090911743899413784034466, −7.16706829018412329478891761561, −6.65299863980579408457836536642, −5.58393014331547726114575068556, −4.91267258189623442618300712247, −3.78438045340965115522055707355, −2.65889593714489926532240422109, −1.64881591307567112670085305000, 1.51842989733061180236391651852, 3.16649378421366817175115481493, 4.20712617639826792368601817554, 4.70356903545416360686776974928, 5.44166710902342945117136858834, 6.21028317959422507123164569761, 7.51512761469919996600133553583, 8.144522968151241496210575590254, 9.199345189460336447250142584786, 10.03946438060623581981675063067

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