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Properties

Label 2-2511-279.194-c0-0-1
Degree $2$
Conductor $2511$
Sign $0.210 + 0.977i$
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  + (−0.336 − 1.58i)2-s + (−1.47 + 0.658i)4-s + (0.866 + 0.5i)5-s + (0.587 + 0.809i)8-s + (0.5 − 1.53i)10-s + (0.994 + 0.104i)11-s + (0.363 + 0.5i)17-s + (−0.309 + 0.951i)19-s + (−1.60 − 0.169i)20-s + (−0.169 − 1.60i)22-s + (1.60 − 0.169i)23-s + (0.207 + 0.978i)29-s + (−0.913 + 0.406i)31-s + (0.866 + 0.499i)32-s + (0.669 − 0.743i)34-s + ⋯
L(s)  = 1  + (−0.336 − 1.58i)2-s + (−1.47 + 0.658i)4-s + (0.866 + 0.5i)5-s + (0.587 + 0.809i)8-s + (0.5 − 1.53i)10-s + (0.994 + 0.104i)11-s + (0.363 + 0.5i)17-s + (−0.309 + 0.951i)19-s + (−1.60 − 0.169i)20-s + (−0.169 − 1.60i)22-s + (1.60 − 0.169i)23-s + (0.207 + 0.978i)29-s + (−0.913 + 0.406i)31-s + (0.866 + 0.499i)32-s + (0.669 − 0.743i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $0.210 + 0.977i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (2240, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ 0.210 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.167951025\)
\(L(\frac12)\) \(\approx\) \(1.167951025\)
\(L(1)\) 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 \( 1 + (0.913 - 0.406i)T \)
good2 \( 1 + (0.336 + 1.58i)T + (-0.913 + 0.406i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.978 - 0.207i)T^{2} \)
11 \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \)
13 \( 1 + (0.913 + 0.406i)T^{2} \)
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 \( 1 + (-1.60 + 0.169i)T + (0.978 - 0.207i)T^{2} \)
29 \( 1 + (-0.207 - 0.978i)T + (-0.913 + 0.406i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (0.459 - 0.413i)T + (0.104 - 0.994i)T^{2} \)
43 \( 1 + (-1.58 + 0.336i)T + (0.913 - 0.406i)T^{2} \)
47 \( 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.128 + 0.604i)T + (-0.913 - 0.406i)T^{2} \)
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 \( 1 + (0.669 + 0.743i)T^{2} \)
83 \( 1 + (0.207 + 0.978i)T + (-0.913 + 0.406i)T^{2} \)
89 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.978 - 0.207i)T^{2} \)
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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

−8.972735019313173290779034915119, −8.835899094791882113382696904342, −7.46060316327219561146550080142, −6.59013404677526847867844410597, −5.82020357654657404760822235087, −4.71860600829365060690909032869, −3.69076297283976613091349041927, −3.08558502188884570584082486154, −1.98511176707554853478033556406, −1.32922230211001428314494733426, 1.05042155892349712170482404975, 2.51261294863026580597427953908, 3.96749603405219667334993880083, 4.97516342060640178112114756575, 5.54384179370412865655062024538, 6.21271631144975539220181503135, 7.10111491549331983460004019372, 7.41889347611213734219217428297, 8.783127808388333672682154085433, 8.927723372760534124089367731234

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