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Properties

Label 2-1260-7.2-c1-0-12
Degree $2$
Conductor $1260$
Sign $-0.868 + 0.495i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + (−1.16 − 2.37i)7-s + (−0.661 − 1.14i)11-s + 0.323·13-s + (−3.13 − 5.43i)17-s + (−1.97 + 3.42i)19-s + (−1.81 + 3.14i)23-s + (−0.499 − 0.866i)25-s − 5.23·29-s + (0.5 + 0.866i)31-s + (−2.63 − 0.182i)35-s + (2.31 − 4.01i)37-s − 8.58·41-s + 5.27·43-s + (−1.95 + 3.38i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.439 − 0.898i)7-s + (−0.199 − 0.345i)11-s + 0.0896·13-s + (−0.761 − 1.31i)17-s + (−0.453 + 0.785i)19-s + (−0.378 + 0.655i)23-s + (−0.0999 − 0.173i)25-s − 0.972·29-s + (0.0898 + 0.155i)31-s + (−0.446 − 0.0308i)35-s + (0.380 − 0.659i)37-s − 1.34·41-s + 0.805·43-s + (−0.285 + 0.494i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.868 + 0.495i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.868 + 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7864392267\)
\(L(\frac12)\) \(\approx\) \(0.7864392267\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.16 + 2.37i)T \)
good11 \( 1 + (0.661 + 1.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.323T + 13T^{2} \)
17 \( 1 + (3.13 + 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.97 - 3.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.81 - 3.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.23T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.31 + 4.01i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.58T + 41T^{2} \)
43 \( 1 - 5.27T + 43T^{2} \)
47 \( 1 + (1.95 - 3.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.27 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.66 + 6.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.47 + 2.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.161 + 0.279i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.67T + 71T^{2} \)
73 \( 1 + (-1.68 - 2.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.25 - 9.10i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.92T + 83T^{2} \)
89 \( 1 + (-6.94 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.60T + 97T^{2} \)
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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

−9.509408989013920917473821262375, −8.553696476089271954727913269355, −7.68390517146659782518591742471, −6.90983738180829522523521904082, −6.02803409734051968045760976740, −5.07989105845013483136967306465, −4.11952161600281470905889368445, −3.19847423502928262805650442327, −1.81852406919958799205090834702, −0.30857671418426030382372427048, 1.91658902448448198267545590123, 2.76395632333989303178100066665, 3.94775411904096963295487524474, 4.97953937938728886460590077372, 6.09110073577950951668683102989, 6.49182226413315489862249372220, 7.58205396028929432337520492242, 8.575207245614028731845910773731, 9.122008453997594560627697517375, 10.06769289077520078930159882222

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