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Properties

Label 2-2511-279.221-c0-0-0
Degree $2$
Conductor $2511$
Sign $-0.136 - 0.990i$
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  + (1.20 + 1.08i)2-s + (0.169 + 1.60i)4-s + (0.866 + 0.5i)5-s + (−0.587 + 0.809i)8-s + (0.499 + 1.53i)10-s + (0.406 − 0.913i)11-s + (−0.363 + 0.5i)17-s + (−0.309 − 0.951i)19-s + (−0.658 + 1.47i)20-s + (1.47 − 0.658i)22-s + (0.658 + 1.47i)23-s + (−0.743 − 0.669i)29-s + (0.104 + 0.994i)31-s + (0.866 + 0.499i)32-s + (−0.978 + 0.207i)34-s + ⋯
L(s)  = 1  + (1.20 + 1.08i)2-s + (0.169 + 1.60i)4-s + (0.866 + 0.5i)5-s + (−0.587 + 0.809i)8-s + (0.499 + 1.53i)10-s + (0.406 − 0.913i)11-s + (−0.363 + 0.5i)17-s + (−0.309 − 0.951i)19-s + (−0.658 + 1.47i)20-s + (1.47 − 0.658i)22-s + (0.658 + 1.47i)23-s + (−0.743 − 0.669i)29-s + (0.104 + 0.994i)31-s + (0.866 + 0.499i)32-s + (−0.978 + 0.207i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $-0.136 - 0.990i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (1430, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ -0.136 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.657401851\)
\(L(\frac12)\) \(\approx\) \(2.657401851\)
\(L(1)\) 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 \( 1 + (-0.104 - 0.994i)T \)
good2 \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.669 + 0.743i)T^{2} \)
11 \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \)
13 \( 1 + (-0.104 + 0.994i)T^{2} \)
17 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.658 - 1.47i)T + (-0.669 + 0.743i)T^{2} \)
29 \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (-0.128 + 0.604i)T + (-0.913 - 0.406i)T^{2} \)
43 \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \)
47 \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.459 - 0.413i)T + (0.104 - 0.994i)T^{2} \)
61 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.978 - 0.207i)T^{2} \)
83 \( 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2} \)
89 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.669 + 0.743i)T^{2} \)
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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

−9.118805464780750815634266379864, −8.379030719966033835618038682177, −7.45197406121658945036643870988, −6.63178015848689071027581716184, −6.31188983055810230585334160756, −5.43826010562547603409569790747, −4.88891583001358419794709115376, −3.69165015583431688763133086421, −3.14701063145524172820552410069, −1.79933881429398395873494766095, 1.49586077182512101676992389141, 2.10877022914572493368198659318, 3.12865403312263666172176204533, 4.15224067347885261452319491171, 4.78988903453332723984009408196, 5.47066069923068988073390809399, 6.26913599148789933973554681421, 7.11368489815999365270699160079, 8.288803583160955183872761434292, 9.217309660367079169494075242611

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