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Properties

Label 2-2511-279.184-c0-0-0
Degree $2$
Conductor $2511$
Sign $-0.0468 - 0.998i$
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  + (0.978 + 0.207i)4-s + (−1.08 + 1.20i)7-s + (−1.89 + 0.198i)13-s + (0.913 + 0.406i)16-s + (0.5 + 0.363i)19-s + (0.5 + 0.866i)25-s + (−1.30 + 0.951i)28-s + (0.104 + 0.994i)31-s + 1.17i·37-s + (1.16 + 0.122i)43-s + (−0.169 − 1.60i)49-s + (−1.89 − 0.198i)52-s + (−1.64 + 0.951i)61-s + (0.809 + 0.587i)64-s + (0.309 − 0.535i)67-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)4-s + (−1.08 + 1.20i)7-s + (−1.89 + 0.198i)13-s + (0.913 + 0.406i)16-s + (0.5 + 0.363i)19-s + (0.5 + 0.866i)25-s + (−1.30 + 0.951i)28-s + (0.104 + 0.994i)31-s + 1.17i·37-s + (1.16 + 0.122i)43-s + (−0.169 − 1.60i)49-s + (−1.89 − 0.198i)52-s + (−1.64 + 0.951i)61-s + (0.809 + 0.587i)64-s + (0.309 − 0.535i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0468 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0468 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $-0.0468 - 0.998i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (1486, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ -0.0468 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.119249447\)
\(L(\frac12)\) \(\approx\) \(1.119249447\)
\(L(1)\) 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 \( 1 + (-0.978 - 0.207i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \)
11 \( 1 + (0.104 + 0.994i)T^{2} \)
13 \( 1 + (1.89 - 0.198i)T + (0.978 - 0.207i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.104 - 0.994i)T^{2} \)
29 \( 1 + (0.978 + 0.207i)T^{2} \)
37 \( 1 - 1.17iT - T^{2} \)
41 \( 1 + (0.669 - 0.743i)T^{2} \)
43 \( 1 + (-1.16 - 0.122i)T + (0.978 + 0.207i)T^{2} \)
47 \( 1 + (0.669 + 0.743i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.978 + 0.207i)T^{2} \)
61 \( 1 + (1.64 - 0.951i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.395 + 1.86i)T + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.978 + 0.207i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.413 + 0.459i)T + (-0.104 - 0.994i)T^{2} \)
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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

−9.361231355355554209419354563352, −8.605507275698777128279095000442, −7.51825075257686241128973474020, −7.08389039464391873566203960637, −6.22288637896785163285983704365, −5.57352334727635101962691215928, −4.66747733335834489900303958839, −3.14015821884673721257069625739, −2.85284573871230746111793493028, −1.79663648387671285289926823738, 0.67990509586643149079029119841, 2.30480184819980589909219943304, 2.98326162407261817078123765707, 4.02189294048254884685964351091, 4.97532613985530790914899489664, 5.99424940914104020415765840020, 6.69851197774734796164991717098, 7.46648779002589860636584318236, 7.62173849766526559939529778620, 9.126897760850256804812121590327

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