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Properties

Label 2-2925-325.38-c0-0-1
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  + (0.266 + 1.68i)2-s + (−1.81 + 0.589i)4-s + (0.996 − 0.0784i)5-s + (−0.702 − 1.37i)8-s + (0.398 + 1.65i)10-s + (−0.893 + 1.23i)11-s + (0.987 + 0.156i)13-s + (0.592 − 0.430i)16-s + (−1.76 + 0.730i)20-s + (−2.31 − 1.17i)22-s + (0.987 − 0.156i)25-s + 1.70i·26-s + (−0.211 − 0.211i)32-s + (−0.809 − 1.32i)40-s + (1.14 + 1.57i)41-s + ⋯
L(s)  = 1  + (0.266 + 1.68i)2-s + (−1.81 + 0.589i)4-s + (0.996 − 0.0784i)5-s + (−0.702 − 1.37i)8-s + (0.398 + 1.65i)10-s + (−0.893 + 1.23i)11-s + (0.987 + 0.156i)13-s + (0.592 − 0.430i)16-s + (−1.76 + 0.730i)20-s + (−2.31 − 1.17i)22-s + (0.987 − 0.156i)25-s + 1.70i·26-s + (−0.211 − 0.211i)32-s + (−0.809 − 1.32i)40-s + (1.14 + 1.57i)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: $\chi_{2925} (2638, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.465536245\)
\(L(\frac12)\) \(\approx\) \(1.465536245\)
\(L(1)\) 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 \( 1 + (-0.996 + 0.0784i)T \)
13 \( 1 + (-0.987 - 0.156i)T \)
good2 \( 1 + (-0.266 - 1.68i)T + (-0.951 + 0.309i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.893 - 1.23i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (-1.14 - 1.57i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
47 \( 1 + (0.882 - 1.73i)T + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.126 - 0.0922i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (-1.23 + 0.401i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.474 + 0.931i)T + (-0.587 + 0.809i)T^{2} \)
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

−9.222348531168015146566764080928, −8.239015703416699143117242786982, −7.75513041174011534240816217315, −6.87606441913337906396959413409, −6.27516587357763812093711453373, −5.66903244527627687117119070178, −4.86168610048569617597683118180, −4.32569602538809484974656711690, −2.92869080234897174969045725155, −1.68122460916283001558188029378, 0.883679325545636970733562282563, 1.95576192413778319616697490106, 2.82346123145537966015535968000, 3.45703684298056589196552978663, 4.42639783921567141696683508341, 5.53466427428011698772245546257, 5.80352030273076974318829638046, 6.99256541701485517613111776053, 8.257602958594341074620619873422, 8.833923322500338164342519913546

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