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Properties

Label 2-45e2-225.11-c0-0-0
Degree $2$
Conductor $2025$
Sign $0.632 + 0.774i$
Analytic cond. $1.01060$
Root an. cond. $1.00528$
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  + (−1.60 + 0.169i)2-s + (1.58 − 0.336i)4-s + (0.994 + 0.104i)5-s + (0.809 − 1.40i)7-s + (−0.951 + 0.309i)8-s − 1.61·10-s + (−0.994 + 0.104i)11-s + (−1.06 + 2.39i)14-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (1.60 − 0.169i)20-s + (1.58 − 0.336i)22-s + (0.406 − 0.913i)23-s + (0.978 + 0.207i)25-s + (0.809 − 2.48i)28-s + (−0.459 + 0.413i)29-s + ⋯
L(s)  = 1  + (−1.60 + 0.169i)2-s + (1.58 − 0.336i)4-s + (0.994 + 0.104i)5-s + (0.809 − 1.40i)7-s + (−0.951 + 0.309i)8-s − 1.61·10-s + (−0.994 + 0.104i)11-s + (−1.06 + 2.39i)14-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (1.60 − 0.169i)20-s + (1.58 − 0.336i)22-s + (0.406 − 0.913i)23-s + (0.978 + 0.207i)25-s + (0.809 − 2.48i)28-s + (−0.459 + 0.413i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.632 + 0.774i$
Analytic conductor: \(1.01060\)
Root analytic conductor: \(1.00528\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :0),\ 0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6655913713\)
\(L(\frac12)\) \(\approx\) \(0.6655913713\)
\(L(1)\) 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 \( 1 + (-0.994 - 0.104i)T \)
good2 \( 1 + (1.60 - 0.169i)T + (0.978 - 0.207i)T^{2} \)
7 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.994 - 0.104i)T + (0.978 - 0.207i)T^{2} \)
13 \( 1 + (-0.978 - 0.207i)T^{2} \)
17 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.406 + 0.913i)T + (-0.669 - 0.743i)T^{2} \)
29 \( 1 + (0.459 - 0.413i)T + (0.104 - 0.994i)T^{2} \)
31 \( 1 + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.994 + 0.104i)T + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.459 + 0.413i)T + (0.104 - 0.994i)T^{2} \)
53 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
67 \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \)
71 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.413 + 0.459i)T + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (-0.336 + 1.58i)T + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \)
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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

−9.229876693606574112154600664719, −8.472301573506011191934382791032, −7.72053122535061740126533931175, −7.17170650626240991410836869070, −6.48799695981347312557962895563, −5.28873624466954495649023696394, −4.52105560304018028912412213731, −2.93557197981825751611825195020, −1.86423148680276174403585449787, −0.861110488983252114048255101583, 1.51236131870347942395225268345, 2.12103829880132840031529713477, 3.07542198747582581182275132850, 4.93627177559431721313640695200, 5.62261181808532735778262774486, 6.28916093615378962212520272344, 7.63431281448436112547891472955, 8.028195089022224065604058132620, 8.758189752992025693713511626794, 9.423484722220268966347829207309

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