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Properties

Label 2-2925-325.103-c0-0-0
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  + (−1.15 + 0.589i)2-s + (0.403 − 0.555i)4-s + (−0.522 + 0.852i)5-s + (0.0635 − 0.401i)8-s + (0.101 − 1.29i)10-s + (−1.84 − 0.600i)11-s + (−0.453 + 0.891i)13-s + (0.375 + 1.15i)16-s + (0.262 + 0.634i)20-s + (2.49 − 0.395i)22-s + (−0.453 − 0.891i)25-s − 1.29i·26-s + (−0.828 − 0.828i)32-s + (0.309 + 0.263i)40-s + (1.89 − 0.616i)41-s + ⋯
L(s)  = 1  + (−1.15 + 0.589i)2-s + (0.403 − 0.555i)4-s + (−0.522 + 0.852i)5-s + (0.0635 − 0.401i)8-s + (0.101 − 1.29i)10-s + (−1.84 − 0.600i)11-s + (−0.453 + 0.891i)13-s + (0.375 + 1.15i)16-s + (0.262 + 0.634i)20-s + (2.49 − 0.395i)22-s + (−0.453 − 0.891i)25-s − 1.29i·26-s + (−0.828 − 0.828i)32-s + (0.309 + 0.263i)40-s + (1.89 − 0.616i)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: $\chi_{2925} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.770 + 0.637i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2342062993\)
\(L(\frac12)\) \(\approx\) \(0.2342062993\)
\(L(1)\) 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 \( 1 + (0.522 - 0.852i)T \)
13 \( 1 + (0.453 - 0.891i)T \)
good2 \( 1 + (1.15 - 0.589i)T + (0.587 - 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (1.84 + 0.600i)T + (0.809 + 0.587i)T^{2} \)
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 \( 1 + (-1.89 + 0.616i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
47 \( 1 + (-0.311 - 1.96i)T + (-0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.526 + 1.62i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (-0.274 + 0.377i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.237 - 1.50i)T + (-0.951 - 0.309i)T^{2} \)
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

−8.653705208398103844995070510148, −8.013145094949396523647199108180, −7.53895754809940179673999679585, −6.87697077287725085847042732665, −6.13554128109802354477725785964, −5.14855539735038886990286402454, −4.06607037118804181113217997623, −3.09967917198494613909819549856, −2.12884012625310415233096025596, −0.25658203818699561909188108663, 1.01125526470503936186658392915, 2.29454077318859968141005737835, 3.01162319870462139744830241152, 4.44073592835420645836076116253, 5.13185550975606116101935492956, 5.75613881698626340871755013116, 7.39563234383320467782434038450, 7.70125703990520540133457555877, 8.338614196613272599432614580751, 9.001186432876158288682574766401

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