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Properties

Label 2-3225-3225.1511-c0-0-1
Degree $2$
Conductor $3225$
Sign $0.489 - 0.871i$
Analytic cond. $1.60948$
Root an. cond. $1.26865$
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  + (0.587 + 0.809i)2-s + (0.669 + 0.743i)3-s + (0.207 − 0.978i)5-s + (−0.207 + 0.978i)6-s + (0.5 + 0.866i)7-s + (0.951 − 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.913 − 0.406i)10-s + (−0.587 − 0.809i)11-s + (0.0646 − 0.614i)13-s + (−0.406 + 0.913i)14-s + (0.866 − 0.5i)15-s + (0.809 + 0.587i)16-s + (−0.866 + 0.5i)18-s + (−0.309 + 0.951i)21-s + (0.309 − 0.951i)22-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.669 + 0.743i)3-s + (0.207 − 0.978i)5-s + (−0.207 + 0.978i)6-s + (0.5 + 0.866i)7-s + (0.951 − 0.309i)8-s + (−0.104 + 0.994i)9-s + (0.913 − 0.406i)10-s + (−0.587 − 0.809i)11-s + (0.0646 − 0.614i)13-s + (−0.406 + 0.913i)14-s + (0.866 − 0.5i)15-s + (0.809 + 0.587i)16-s + (−0.866 + 0.5i)18-s + (−0.309 + 0.951i)21-s + (0.309 − 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3225\)    =    \(3 \cdot 5^{2} \cdot 43\)
Sign: $0.489 - 0.871i$
Analytic conductor: \(1.60948\)
Root analytic conductor: \(1.26865\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3225} (1511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3225,\ (\ :0),\ 0.489 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.485252816\)
\(L(\frac12)\) \(\approx\) \(2.485252816\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (-0.207 + 0.978i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.0646 + 0.614i)T + (-0.978 - 0.207i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.913 + 0.406i)T^{2} \)
23 \( 1 + (0.614 - 0.0646i)T + (0.978 - 0.207i)T^{2} \)
29 \( 1 + (-0.336 - 1.58i)T + (-0.913 + 0.406i)T^{2} \)
31 \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \)
37 \( 1 + (0.564 - 0.251i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.104 - 0.994i)T^{2} \)
59 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.669 + 0.743i)T^{2} \)
67 \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \)
71 \( 1 + (0.336 + 1.58i)T + (-0.913 + 0.406i)T^{2} \)
73 \( 1 + (1.47 + 0.658i)T + (0.669 + 0.743i)T^{2} \)
79 \( 1 + (0.913 - 0.406i)T^{2} \)
83 \( 1 + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \)
97 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−8.721710307036223310493311980402, −8.246867536518256346858799939319, −7.73077977451807517553744322075, −6.50286400470710610015757193890, −5.54324399835876690148789053103, −5.27220559716776361214781259347, −4.64520631136279144106291635478, −3.65961065217198175712324997694, −2.60336158720239852588435311550, −1.47786613819193038268893097432, 1.48813568487221750608469186029, 2.33786527994402287916765911710, 2.85194436810191269680459562342, 4.02808754225934735268656128593, 4.31513987450987510407229526326, 5.68537146346325865668093544606, 6.74272348446094570195429612813, 7.24398133124182877624232658148, 7.84737350904007870464232995999, 8.453303097874445425378029997299

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