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Properties

Label 2-2565-2565.664-c0-0-3
Degree $2$
Conductor $2565$
Sign $-0.784 - 0.620i$
Analytic cond. $1.28010$
Root an. cond. $1.13141$
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  + (1.38 + 1.16i)2-s + (−0.906 − 0.422i)3-s + (0.396 + 2.25i)4-s + (0.939 + 0.342i)5-s + (−0.766 − 1.64i)6-s + (−1.16 + 2.01i)8-s + (0.642 + 0.766i)9-s + (0.906 + 1.56i)10-s + (−0.642 + 0.233i)11-s + (0.591 − 2.20i)12-s + (−1.52 + 1.28i)13-s + (−0.707 − 0.707i)15-s + (−1.82 + 0.662i)16-s + 1.81i·18-s + (0.5 − 0.866i)19-s + (−0.396 + 2.25i)20-s + ⋯
L(s)  = 1  + (1.38 + 1.16i)2-s + (−0.906 − 0.422i)3-s + (0.396 + 2.25i)4-s + (0.939 + 0.342i)5-s + (−0.766 − 1.64i)6-s + (−1.16 + 2.01i)8-s + (0.642 + 0.766i)9-s + (0.906 + 1.56i)10-s + (−0.642 + 0.233i)11-s + (0.591 − 2.20i)12-s + (−1.52 + 1.28i)13-s + (−0.707 − 0.707i)15-s + (−1.82 + 0.662i)16-s + 1.81i·18-s + (0.5 − 0.866i)19-s + (−0.396 + 2.25i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2565 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2565 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2565\)    =    \(3^{3} \cdot 5 \cdot 19\)
Sign: $-0.784 - 0.620i$
Analytic conductor: \(1.28010\)
Root analytic conductor: \(1.13141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2565} (664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2565,\ (\ :0),\ -0.784 - 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.017136514\)
\(L(\frac12)\) \(\approx\) \(2.017136514\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.906 + 0.422i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.38 - 1.16i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.642 - 0.233i)T + (0.766 - 0.642i)T^{2} \)
13 \( 1 + (1.52 - 1.28i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.939 - 0.342i)T^{2} \)
37 \( 1 + (-0.422 - 0.731i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 - 1.63T + T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.47 + 1.24i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.486 - 0.177i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.415851830444707465603249291492, −8.161867342453939977857392797628, −7.21795808404961000936925587955, −6.95399420590398260588057978991, −6.30780472585474632799290805718, −5.39735414720442548495095118129, −4.99826171125501772925583760137, −4.31961734367023049749362543575, −2.86355940305672815992921063292, −2.05801718759323415713169708839, 0.933776080285739571961483271423, 2.22559154254152848361003385199, 3.01993646813311245175178401938, 4.05582827229644002065188708875, 4.98204287320928696192119267126, 5.43481176711944204703703498908, 5.79803205547556929925799321978, 6.82754120215062198490447680821, 7.950619807434767837273467894481, 9.381478087972110741647919666012

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