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Properties

Label 2-2511-279.61-c0-0-14
Degree $2$
Conductor $2511$
Sign $0.766 + 0.642i$
Analytic cond. $1.25315$
Root an. cond. $1.11944$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + 0.999·10-s + (0.499 + 0.866i)14-s + (0.5 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)31-s − 0.999·35-s + (0.5 − 0.866i)38-s + (0.500 + 0.866i)40-s + (−0.5 − 0.866i)41-s + (1 − 1.73i)47-s + (−0.499 + 0.866i)56-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + 0.999·10-s + (0.499 + 0.866i)14-s + (0.5 − 0.866i)16-s − 19-s + (−0.5 − 0.866i)31-s − 0.999·35-s + (0.5 − 0.866i)38-s + (0.500 + 0.866i)40-s + (−0.5 − 0.866i)41-s + (1 − 1.73i)47-s + (−0.499 + 0.866i)56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (1270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6795395114\)
\(L(\frac12)\) \(\approx\) \(0.6795395114\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.736323713411248440203419789786, −8.198478000487596485289428798442, −7.58846462201935680829343645074, −6.93931555965084014063122329964, −6.09895773373300085959982092023, −5.14344558549566103168630272916, −4.29176497678808437394661167134, −3.55015815251004854649158269162, −2.08705053634266367403091249959, −0.53112701819777788176625427430, 1.48692283923433881488862561605, 2.49390572277752623463453504994, 3.11426226292839122231162671076, 4.19636261136625581228050107981, 5.33773211109242105518335590082, 6.14274630484919962006846332736, 6.89020714323617654750340770777, 7.81902750449512159214704298116, 8.676405056856754393515236625753, 9.145501754859929174447904801090

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