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Properties

Label 2-325-5.4-c5-0-53
Degree $2$
Conductor $325$
Sign $-0.447 - 0.894i$
Analytic cond. $52.1247$
Root an. cond. $7.21974$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.60i·2-s + 25.9i·3-s − 60.1·4-s − 248.·6-s − 181. i·7-s − 270. i·8-s − 428.·9-s + 512.·11-s − 1.56e3i·12-s + 169i·13-s + 1.74e3·14-s + 672.·16-s − 1.66e3i·17-s − 4.11e3i·18-s + 464.·19-s + ⋯
L(s)  = 1  + 1.69i·2-s + 1.66i·3-s − 1.88·4-s − 2.82·6-s − 1.40i·7-s − 1.49i·8-s − 1.76·9-s + 1.27·11-s − 3.12i·12-s + 0.277i·13-s + 2.37·14-s + 0.656·16-s − 1.39i·17-s − 2.99i·18-s + 0.294·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(52.1247\)
Root analytic conductor: \(7.21974\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :5/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.559817433\)
\(L(\frac12)\) \(\approx\) \(1.559817433\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 - 169iT \)
good2 \( 1 - 9.60iT - 32T^{2} \)
3 \( 1 - 25.9iT - 243T^{2} \)
7 \( 1 + 181. iT - 1.68e4T^{2} \)
11 \( 1 - 512.T + 1.61e5T^{2} \)
17 \( 1 + 1.66e3iT - 1.41e6T^{2} \)
19 \( 1 - 464.T + 2.47e6T^{2} \)
23 \( 1 + 2.09e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.93e3T + 2.05e7T^{2} \)
31 \( 1 - 8.39e3T + 2.86e7T^{2} \)
37 \( 1 - 8.90e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.25e3T + 1.15e8T^{2} \)
43 \( 1 + 5.95e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.07e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.93e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.94e3T + 7.14e8T^{2} \)
61 \( 1 + 4.14e4T + 8.44e8T^{2} \)
67 \( 1 + 1.85e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.03e4T + 1.80e9T^{2} \)
73 \( 1 + 7.86e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.17e4T + 3.07e9T^{2} \)
83 \( 1 - 208. iT - 3.93e9T^{2} \)
89 \( 1 - 4.35e4T + 5.58e9T^{2} \)
97 \( 1 + 1.52e4iT - 8.58e9T^{2} \)
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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

−10.80076137437969879670734338116, −9.676248607722121466522662707605, −9.307825465624314241780469731357, −8.193440158638402609765536147099, −7.09915689264292416481870545801, −6.30931917808127686327597705953, −4.94456368766598374432465716330, −4.42463094303585851322394886075, −3.51620124747826154007512723749, −0.49986923259793733657225560343, 1.03658712336380190183131111077, 1.79114366627448945750490698218, 2.63008633393877599448307514883, 3.83967728628371824602506533693, 5.59928607361801479129843137393, 6.45924533345049344114644287252, 7.87364698156272260791929868705, 8.820113150144340549271879017196, 9.476142008553620246253101588813, 10.89146354922169951510757112346

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