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Properties

Label 2-2565-2565.2374-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  + (0.0302 − 0.171i)2-s + (−0.0871 − 0.996i)3-s + (0.911 + 0.331i)4-s + (−0.766 + 0.642i)5-s + (−0.173 − 0.0151i)6-s + (0.171 − 0.297i)8-s + (−0.984 + 0.173i)9-s + (0.0871 + 0.150i)10-s + (0.984 + 0.826i)11-s + (0.250 − 0.936i)12-s + (−0.199 − 1.12i)13-s + (0.707 + 0.707i)15-s + (0.696 + 0.584i)16-s + 0.174i·18-s + (0.5 − 0.866i)19-s + (−0.911 + 0.331i)20-s + ⋯
L(s)  = 1  + (0.0302 − 0.171i)2-s + (−0.0871 − 0.996i)3-s + (0.911 + 0.331i)4-s + (−0.766 + 0.642i)5-s + (−0.173 − 0.0151i)6-s + (0.171 − 0.297i)8-s + (−0.984 + 0.173i)9-s + (0.0871 + 0.150i)10-s + (0.984 + 0.826i)11-s + (0.250 − 0.936i)12-s + (−0.199 − 1.12i)13-s + (0.707 + 0.707i)15-s + (0.696 + 0.584i)16-s + 0.174i·18-s + (0.5 − 0.866i)19-s + (−0.911 + 0.331i)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: $\chi_{2565} (2374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2565,\ (\ :0),\ 0.784 + 0.620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.365526239\)
\(L(\frac12)\) \(\approx\) \(1.365526239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0871 + 0.996i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.0302 + 0.171i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.766 + 0.642i)T^{2} \)
11 \( 1 + (-0.984 - 0.826i)T + (0.173 + 0.984i)T^{2} \)
13 \( 1 + (0.199 + 1.12i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (-0.996 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 - 1.81T + T^{2} \)
59 \( 1 + (-0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.335 + 1.90i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.396 + 0.332i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.755939658072240003353277613690, −7.87908142650361056724100804256, −7.46933239917122118289258727031, −6.76393473057337474246426490473, −6.33240336694197893961295933787, −5.17528335793076033833016597418, −3.94933942123028220542554738698, −3.01644664193487309479265329883, −2.40690778876505476766700825720, −1.12736249943108080296401557415, 1.20643458084646381039756785810, 2.62117748749289172582245719699, 3.82711671830584279139468315128, 4.17730766033855888274475778649, 5.39194817518870746200888227970, 5.88904240462357323450321979880, 6.83647687103152360913214631319, 7.63528426991486610167636254442, 8.536798765066890086700198271796, 9.147182086710326805007802239124

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