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Properties

Label 2-3325-3325.1272-c0-0-1
Degree $2$
Conductor $3325$
Sign $-0.756 - 0.654i$
Analytic cond. $1.65939$
Root an. cond. $1.28817$
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  + (−0.994 − 0.104i)4-s + (−0.309 − 0.951i)5-s + (−0.669 − 0.743i)7-s + (0.207 − 0.978i)9-s + (−1.45 + 0.309i)11-s + (0.978 + 0.207i)16-s + (0.292 − 0.112i)17-s + (−0.104 − 0.994i)19-s + (0.207 + 0.978i)20-s + (0.104 + 0.00547i)23-s + (−0.809 + 0.587i)25-s + (0.587 + 0.809i)28-s + (−0.499 + 0.866i)35-s + (−0.309 + 0.951i)36-s + (−1.09 + 1.09i)43-s + (1.47 − 0.155i)44-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)4-s + (−0.309 − 0.951i)5-s + (−0.669 − 0.743i)7-s + (0.207 − 0.978i)9-s + (−1.45 + 0.309i)11-s + (0.978 + 0.207i)16-s + (0.292 − 0.112i)17-s + (−0.104 − 0.994i)19-s + (0.207 + 0.978i)20-s + (0.104 + 0.00547i)23-s + (−0.809 + 0.587i)25-s + (0.587 + 0.809i)28-s + (−0.499 + 0.866i)35-s + (−0.309 + 0.951i)36-s + (−1.09 + 1.09i)43-s + (1.47 − 0.155i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3325\)    =    \(5^{2} \cdot 7 \cdot 19\)
Sign: $-0.756 - 0.654i$
Analytic conductor: \(1.65939\)
Root analytic conductor: \(1.28817\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3325} (1272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3325,\ (\ :0),\ -0.756 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2077967034\)
\(L(\frac12)\) \(\approx\) \(0.2077967034\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
good2 \( 1 + (0.994 + 0.104i)T^{2} \)
3 \( 1 + (-0.207 + 0.978i)T^{2} \)
11 \( 1 + (1.45 - 0.309i)T + (0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.292 + 0.112i)T + (0.743 - 0.669i)T^{2} \)
23 \( 1 + (-0.104 - 0.00547i)T + (0.994 + 0.104i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.669 + 0.743i)T^{2} \)
37 \( 1 + (0.406 - 0.913i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (1.09 - 1.09i)T - iT^{2} \)
47 \( 1 + (-0.185 + 0.483i)T + (-0.743 - 0.669i)T^{2} \)
53 \( 1 + (0.207 - 0.978i)T^{2} \)
59 \( 1 + (0.104 + 0.994i)T^{2} \)
61 \( 1 + (1.35 - 1.22i)T + (0.104 - 0.994i)T^{2} \)
67 \( 1 + (0.743 - 0.669i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.62 - 1.05i)T + (0.406 + 0.913i)T^{2} \)
79 \( 1 + (0.669 - 0.743i)T^{2} \)
83 \( 1 + (1.24 + 0.196i)T + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (0.104 - 0.994i)T^{2} \)
97 \( 1 + (-0.951 + 0.309i)T^{2} \)
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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

−8.428618694712444346651203134488, −7.70892533688720336639797756214, −6.98245648861004789989698131183, −5.96026822845198797666967423702, −5.10695092220539338852457840523, −4.53188367057677904334289057456, −3.77434336232799117845956327033, −2.90864460989881345819669720600, −1.16110277228804679792471634551, −0.14184138026727531085643550438, 2.08176424582762355489884455304, 3.02053488339789346976236366247, 3.64798213267294354047564774152, 4.77374500067587074562447199022, 5.47182147604027608382740504076, 6.10822743609545962155057578312, 7.19148686339973024664819752621, 8.020413927682145103267500804682, 8.239738778328221795984607707983, 9.295837581700436972026651438607

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