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Properties

Label 2-1260-1260.499-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7609237508\)
\(L(\frac12)\) \(\approx\) \(0.7609237508\)
\(L(1)\) 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

−9.792943128739386561937933465380, −8.949693717962469567784953589874, −8.803774202252009732078253178272, −7.906275736739085378071031982733, −7.18672108008534280345880671456, −5.62584183176806742642517594738, −5.13723730185795925571389087856, −3.86255355668347233751300706722, −2.82687086536661439279922764917, −1.58125098722236226816032540179, 0.918707549892505776501049893952, 2.33821250662494675158540583083, 3.14053318006425320907809437051, 4.35270252152875581853806743661, 6.08140068396568642652600271847, 6.83026847723848053162463416563, 7.32288600453071522221585173184, 8.115233629884421508509022543006, 8.595604449712224635771108269590, 9.733517709600166635968790034880

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