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Properties

Label 2-2511-279.2-c0-0-2
Degree $2$
Conductor $2511$
Sign $0.982 + 0.188i$
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  + (0.251 + 0.564i)2-s + (0.413 − 0.459i)4-s + (0.866 − 0.5i)5-s + (0.951 + 0.309i)8-s + (0.5 + 0.363i)10-s + (0.207 − 0.978i)11-s + (−1.53 − 0.5i)17-s + (0.809 + 0.587i)19-s + (0.128 − 0.604i)20-s + (0.604 − 0.128i)22-s + (−0.128 − 0.604i)23-s + (0.406 + 0.913i)29-s + (−0.669 + 0.743i)31-s + (0.866 − 0.499i)32-s + (−0.104 − 0.994i)34-s + ⋯
L(s)  = 1  + (0.251 + 0.564i)2-s + (0.413 − 0.459i)4-s + (0.866 − 0.5i)5-s + (0.951 + 0.309i)8-s + (0.5 + 0.363i)10-s + (0.207 − 0.978i)11-s + (−1.53 − 0.5i)17-s + (0.809 + 0.587i)19-s + (0.128 − 0.604i)20-s + (0.604 − 0.128i)22-s + (−0.128 − 0.604i)23-s + (0.406 + 0.913i)29-s + (−0.669 + 0.743i)31-s + (0.866 − 0.499i)32-s + (−0.104 − 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $0.982 + 0.188i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ 0.982 + 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.860377281\)
\(L(\frac12)\) \(\approx\) \(1.860377281\)
\(L(1)\) 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 \( 1 + (-0.251 - 0.564i)T + (-0.669 + 0.743i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.913 + 0.406i)T^{2} \)
11 \( 1 + (-0.207 + 0.978i)T + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.669 + 0.743i)T^{2} \)
17 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.128 + 0.604i)T + (-0.913 + 0.406i)T^{2} \)
29 \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (1.60 + 0.169i)T + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.564 + 0.251i)T + (0.669 - 0.743i)T^{2} \)
47 \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.658 - 1.47i)T + (-0.669 - 0.743i)T^{2} \)
61 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \)
89 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.913 + 0.406i)T^{2} \)
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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

−8.953743559645423439707701221509, −8.419141520296187729491144565178, −7.32369599175237784104799221981, −6.62054405058925058050163109269, −6.00268922318601616832795027769, −5.26275920510416846313631086774, −4.69363127370115451987574690730, −3.37860494305724804414302445946, −2.19356718549486169854793714579, −1.25946153606357787320424171069, 1.79730326696357636296116013208, 2.28396258187028366756104944959, 3.28307162450344880870509304596, 4.25221433040362607119052529007, 5.00553152777893011657043634557, 6.28771439466123121050471874769, 6.68427487077827549648757761908, 7.53339687994358372854216220767, 8.280931836306501942143558562414, 9.511522918694291199152687670667

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