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Properties

Label 2-5e2-5.4-c13-0-11
Degree $2$
Conductor $25$
Sign $-0.894 + 0.447i$
Analytic cond. $26.8077$
Root an. cond. $5.17761$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 117. i·2-s + 235. i·3-s − 5.72e3·4-s + 2.78e4·6-s + 2.27e5i·7-s − 2.91e5i·8-s + 1.53e6·9-s − 7.68e5·11-s − 1.34e6i·12-s − 8.24e6i·13-s + 2.68e7·14-s − 8.12e7·16-s − 1.56e8i·17-s − 1.81e8i·18-s + 2.90e8·19-s + ⋯
L(s)  = 1  − 1.30i·2-s + 0.186i·3-s − 0.698·4-s + 0.243·6-s + 0.731i·7-s − 0.392i·8-s + 0.965·9-s − 0.130·11-s − 0.130i·12-s − 0.473i·13-s + 0.953·14-s − 1.21·16-s − 1.56i·17-s − 1.25i·18-s + 1.41·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(26.8077\)
Root analytic conductor: \(5.17761\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :13/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.854946766\)
\(L(\frac12)\) \(\approx\) \(1.854946766\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + 117. iT - 8.19e3T^{2} \)
3 \( 1 - 235. iT - 1.59e6T^{2} \)
7 \( 1 - 2.27e5iT - 9.68e10T^{2} \)
11 \( 1 + 7.68e5T + 3.45e13T^{2} \)
13 \( 1 + 8.24e6iT - 3.02e14T^{2} \)
17 \( 1 + 1.56e8iT - 9.90e15T^{2} \)
19 \( 1 - 2.90e8T + 4.20e16T^{2} \)
23 \( 1 + 7.93e8iT - 5.04e17T^{2} \)
29 \( 1 + 5.40e9T + 1.02e19T^{2} \)
31 \( 1 + 1.83e9T + 2.44e19T^{2} \)
37 \( 1 - 1.16e10iT - 2.43e20T^{2} \)
41 \( 1 + 6.75e9T + 9.25e20T^{2} \)
43 \( 1 + 5.43e10iT - 1.71e21T^{2} \)
47 \( 1 + 8.65e10iT - 5.46e21T^{2} \)
53 \( 1 + 7.13e10iT - 2.60e22T^{2} \)
59 \( 1 + 2.96e11T + 1.04e23T^{2} \)
61 \( 1 - 6.88e11T + 1.61e23T^{2} \)
67 \( 1 + 7.65e11iT - 5.48e23T^{2} \)
71 \( 1 - 4.74e11T + 1.16e24T^{2} \)
73 \( 1 + 4.21e11iT - 1.67e24T^{2} \)
79 \( 1 - 1.91e12T + 4.66e24T^{2} \)
83 \( 1 + 5.01e12iT - 8.87e24T^{2} \)
89 \( 1 + 8.23e12T + 2.19e25T^{2} \)
97 \( 1 - 9.34e12iT - 6.73e25T^{2} \)
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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

−13.55360712395597660042605266018, −12.42808371515737694401814024512, −11.47583877356721729692502583109, −10.15580158852998535847077322078, −9.212973746608777893044574728303, −7.19379533190319719441158525396, −5.09580326825808921568109372491, −3.44012432973947492957532082583, −2.14622852778324230605231478757, −0.62904273297424593624247668729, 1.51470630114450274679922299026, 4.01526730875373665832506845241, 5.65097953579954002450127540363, 7.05599400903144499464353643125, 7.82178215003155234387849796202, 9.545670875188983047494320870838, 11.14808650791294551323872107884, 12.94084470046548052813811735742, 14.05077065915650343965087436781, 15.23315547684687749696089969148

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