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Properties

Label 2-2565-285.104-c0-0-1
Degree $2$
Conductor $2565$
Sign $0.944 + 0.327i$
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  + (−0.0603 − 0.342i)2-s + (0.826 − 0.300i)4-s + (0.939 + 0.342i)5-s + (−0.326 − 0.565i)8-s + (0.0603 − 0.342i)10-s + (0.500 − 0.419i)16-s + (0.326 + 1.85i)17-s + (0.173 + 0.984i)19-s + 0.879·20-s + (−0.939 + 0.342i)23-s + (0.766 + 0.642i)25-s + (0.939 − 1.62i)31-s + (−0.673 − 0.565i)32-s + (0.613 − 0.223i)34-s + (0.326 − 0.118i)38-s + ⋯
L(s)  = 1  + (−0.0603 − 0.342i)2-s + (0.826 − 0.300i)4-s + (0.939 + 0.342i)5-s + (−0.326 − 0.565i)8-s + (0.0603 − 0.342i)10-s + (0.500 − 0.419i)16-s + (0.326 + 1.85i)17-s + (0.173 + 0.984i)19-s + 0.879·20-s + (−0.939 + 0.342i)23-s + (0.766 + 0.642i)25-s + (0.939 − 1.62i)31-s + (−0.673 − 0.565i)32-s + (0.613 − 0.223i)34-s + (0.326 − 0.118i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2565 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2565 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2565\)    =    \(3^{3} \cdot 5 \cdot 19\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(1.28010\)
Root analytic conductor: \(1.13141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2565} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2565,\ (\ :0),\ 0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.705337955\)
\(L(\frac12)\) \(\approx\) \(1.705337955\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
good2 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.939 - 0.342i)T^{2} \)
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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

−9.289647680340892718657292314086, −8.163233882435160883470008150344, −7.58007170250753751247547453578, −6.37050439164157694072843107468, −6.15686814545282114815799354706, −5.42534631191617581573744492681, −4.03113528832412161928840033850, −3.19913360511935632914692612126, −2.08954395142792949552439714110, −1.53482759991775727237543845202, 1.34273241557004524902751479290, 2.56614066740727837225555871236, 3.07010240992553707853905147213, 4.67548675952821889565171222471, 5.22582667225081844701077594945, 6.26471324217872773685277125468, 6.67542909973361580380817201425, 7.57836269155376602810245774838, 8.238675677183775010818901774005, 9.238546475901844285900710063405

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