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Properties

Label 2-2925-13.11-c0-0-0
Degree $2$
Conductor $2925$
Sign $0.846 - 0.533i$
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  + (0.866 + 0.5i)4-s + (−0.133 − 0.5i)7-s + i·13-s + (0.499 + 0.866i)16-s + (0.5 − 0.133i)19-s + (0.133 − 0.5i)28-s + (1 + i)31-s + (−1.36 − 0.366i)37-s + (1.5 + 0.866i)43-s + (0.633 − 0.366i)49-s + (−0.5 + 0.866i)52-s + 0.999i·64-s + (0.5 − 1.86i)67-s + (1.36 − 1.36i)73-s + (0.5 + 0.133i)76-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)4-s + (−0.133 − 0.5i)7-s + i·13-s + (0.499 + 0.866i)16-s + (0.5 − 0.133i)19-s + (0.133 − 0.5i)28-s + (1 + i)31-s + (−1.36 − 0.366i)37-s + (1.5 + 0.866i)43-s + (0.633 − 0.366i)49-s + (−0.5 + 0.866i)52-s + 0.999i·64-s + (0.5 − 1.86i)67-s + (1.36 − 1.36i)73-s + (0.5 + 0.133i)76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.846 - 0.533i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2026, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.846 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.536291316\)
\(L(\frac12)\) \(\approx\) \(1.536291316\)
\(L(1)\) 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 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 + (-0.866 - 0.5i)T^{2} \)
7 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.968412418243387139779450252378, −8.149398920350000985908300724421, −7.39406437280968512491543060201, −6.81159248079131771615249375363, −6.21726017092252145932287880309, −5.13013903807080787706900766460, −4.16009326165230679354799522898, −3.38529206695337851366642570780, −2.45634923613531577089378562167, −1.40124941324008721465109505307, 1.07496769909569274712228116869, 2.35093600868127115097507172037, 2.98168475493905534419563162118, 4.10340289122770403259169187619, 5.42027503229492894702493710454, 5.64751263112407751529431816919, 6.60133685358948335202945302332, 7.31457812433924487993888177744, 8.054884010918693896730284422833, 8.849196545492404422063075134130

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