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Properties

Label 2-2925-325.233-c0-0-3
Degree $2$
Conductor $2925$
Sign $0.770 - 0.637i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
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  + (0.0712 − 0.139i)2-s + (0.573 + 0.789i)4-s + (0.972 + 0.233i)5-s + (0.306 − 0.0484i)8-s + (0.101 − 0.119i)10-s + (−0.993 + 0.322i)11-s + (0.891 − 0.453i)13-s + (−0.286 + 0.881i)16-s + (0.373 + 0.901i)20-s + (−0.0256 + 0.161i)22-s + (0.891 + 0.453i)25-s − 0.156i·26-s + (0.322 + 0.322i)32-s + (0.309 + 0.0243i)40-s + (−1.44 − 0.469i)41-s + ⋯
L(s)  = 1  + (0.0712 − 0.139i)2-s + (0.573 + 0.789i)4-s + (0.972 + 0.233i)5-s + (0.306 − 0.0484i)8-s + (0.101 − 0.119i)10-s + (−0.993 + 0.322i)11-s + (0.891 − 0.453i)13-s + (−0.286 + 0.881i)16-s + (0.373 + 0.901i)20-s + (−0.0256 + 0.161i)22-s + (0.891 + 0.453i)25-s − 0.156i·26-s + (0.322 + 0.322i)32-s + (0.309 + 0.0243i)40-s + (−1.44 − 0.469i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.770 - 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.750044969\)
\(L(\frac12)\) \(\approx\) \(1.750044969\)
\(L(1)\) 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 \( 1 + (-0.972 - 0.233i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
good2 \( 1 + (-0.0712 + 0.139i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.993 - 0.322i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.587 - 0.809i)T^{2} \)
41 \( 1 + (1.44 + 0.469i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
47 \( 1 + (-1.50 - 0.237i)T + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.144 - 0.444i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (1.00 + 1.37i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.96 - 0.311i)T + (0.951 - 0.309i)T^{2} \)
89 \( 1 + (0.570 + 1.75i)T + (-0.809 + 0.587i)T^{2} \)
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

−8.896259083504711396970115808333, −8.281627452447312546695085482677, −7.46331489067866362053950706558, −6.81780271997909887084830564001, −5.96868772777055664947121978148, −5.31559176714666844707879766736, −4.21576919754547112066326864603, −3.18782240357684974110494484529, −2.53768838531156109435436596191, −1.58951643339579238638750269397, 1.17479516411555596817246655180, 2.12060258513175913225124931465, 2.97598624901969284368521850766, 4.32106957144146882200490058795, 5.31211440284782229678208657608, 5.73455335535716244376243110238, 6.46593417909225265992439108449, 7.16004840701057637740841494821, 8.158404099365142527193369304792, 8.925635493732790380083345735132

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