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Properties

Label 2-2511-279.101-c0-0-1
Degree $2$
Conductor $2511$
Sign $-0.278 - 0.960i$
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  + (−0.104 + 0.994i)4-s + (0.564 − 0.251i)7-s + (−1.08 + 1.20i)13-s + (−0.978 − 0.207i)16-s + (−0.5 + 1.53i)19-s + (−0.5 + 0.866i)25-s + (0.190 + 0.587i)28-s + (0.669 − 0.743i)31-s + 0.618·37-s + (0.413 + 0.459i)43-s + (−0.413 + 0.459i)49-s + (−1.08 − 1.20i)52-s + (0.809 − 1.40i)61-s + (0.309 − 0.951i)64-s + (0.809 + 1.40i)67-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)4-s + (0.564 − 0.251i)7-s + (−1.08 + 1.20i)13-s + (−0.978 − 0.207i)16-s + (−0.5 + 1.53i)19-s + (−0.5 + 0.866i)25-s + (0.190 + 0.587i)28-s + (0.669 − 0.743i)31-s + 0.618·37-s + (0.413 + 0.459i)43-s + (−0.413 + 0.459i)49-s + (−1.08 − 1.20i)52-s + (0.809 − 1.40i)61-s + (0.309 − 0.951i)64-s + (0.809 + 1.40i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $-0.278 - 0.960i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ -0.278 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003145272\)
\(L(\frac12)\) \(\approx\) \(1.003145272\)
\(L(1)\) 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 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (0.104 - 0.994i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.564 + 0.251i)T + (0.669 - 0.743i)T^{2} \)
11 \( 1 + (-0.669 + 0.743i)T^{2} \)
13 \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.669 - 0.743i)T^{2} \)
29 \( 1 + (0.104 - 0.994i)T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (-0.913 + 0.406i)T^{2} \)
43 \( 1 + (-0.413 - 0.459i)T + (-0.104 + 0.994i)T^{2} \)
47 \( 1 + (-0.913 - 0.406i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.104 + 0.994i)T^{2} \)
61 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.169 - 1.60i)T + (-0.978 + 0.207i)T^{2} \)
83 \( 1 + (0.104 - 0.994i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (1.47 - 0.658i)T + (0.669 - 0.743i)T^{2} \)
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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

−9.414672378132280520497990081156, −8.319112432850178832289165360602, −7.899290423823671308218251398888, −7.17608794713419402076182468667, −6.40503475171987412924438545085, −5.30426513651724435891572426631, −4.32497179845622968446994134745, −3.91076476273912134695767796039, −2.66490427430029831017712753816, −1.74330176574112931706888848268, 0.65030611362237814476869179051, 2.09501948735577143308945237285, 2.85822488094126075142506885693, 4.39492422259887843484813627366, 4.97028529316130858024976896025, 5.63891942972321093347582216278, 6.51897653875975072507969505070, 7.30281521422166197455958818635, 8.209455789666020802476783312939, 8.881061307607022340230116034937

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