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Properties

Label 2-2925-325.12-c0-0-2
Degree $2$
Conductor $2925$
Sign $0.992 + 0.125i$
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  + (0.461 − 0.0730i)2-s + (−0.743 + 0.241i)4-s + (0.760 + 0.649i)5-s + (−0.741 + 0.377i)8-s + (0.398 + 0.243i)10-s + (1.17 − 1.61i)11-s + (0.156 − 0.987i)13-s + (0.318 − 0.231i)16-s + (−0.722 − 0.299i)20-s + (0.422 − 0.829i)22-s + (0.156 + 0.987i)25-s − 0.466i·26-s + (0.718 − 0.718i)32-s + (−0.809 − 0.194i)40-s + (0.614 + 0.845i)41-s + ⋯
L(s)  = 1  + (0.461 − 0.0730i)2-s + (−0.743 + 0.241i)4-s + (0.760 + 0.649i)5-s + (−0.741 + 0.377i)8-s + (0.398 + 0.243i)10-s + (1.17 − 1.61i)11-s + (0.156 − 0.987i)13-s + (0.318 − 0.231i)16-s + (−0.722 − 0.299i)20-s + (0.422 − 0.829i)22-s + (0.156 + 0.987i)25-s − 0.466i·26-s + (0.718 − 0.718i)32-s + (−0.809 − 0.194i)40-s + (0.614 + 0.845i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547883863\)
\(L(\frac12)\) \(\approx\) \(1.547883863\)
\(L(1)\) 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 \( 1 + (-0.760 - 0.649i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
good2 \( 1 + (-0.461 + 0.0730i)T + (0.951 - 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-1.17 + 1.61i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (-0.614 - 0.845i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
47 \( 1 + (0.931 + 0.474i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (1.05 - 0.763i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.587 + 0.809i)T^{2} \)
71 \( 1 + (0.149 - 0.0484i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (1.73 - 0.882i)T + (0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.619 - 0.449i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.587 + 0.809i)T^{2} \)
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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

−8.961486462189422893324194648898, −8.322413109019182277946655759904, −7.43105972100495952387422834840, −6.28750062040265683329061042798, −5.91281224208875744364619952161, −5.19270135902047218808564618115, −4.02346519353991905808652398370, −3.35825025233521115967572611388, −2.65944612810371298314360444653, −1.04527719821599804975163899398, 1.29671487828131831016704625562, 2.16412144468817737900852322023, 3.68306056318865630328052550779, 4.52395221138408306769255336898, 4.79652736746631783574378609889, 5.91556867666782044882586119490, 6.46723431298484164456481756456, 7.29180566699120107061966367360, 8.436523834748319099849060357880, 9.179479338408554206029792016780

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