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Properties

Label 2-325-5.4-c5-0-54
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  + 8.11i·2-s − 3.22i·3-s − 33.8·4-s + 26.2·6-s + 20.7i·7-s − 15.2i·8-s + 232.·9-s + 134.·11-s + 109. i·12-s + 169i·13-s − 168.·14-s − 960.·16-s − 2.19e3i·17-s + 1.88e3i·18-s + 1.98e3·19-s + ⋯
L(s)  = 1  + 1.43i·2-s − 0.207i·3-s − 1.05·4-s + 0.297·6-s + 0.159i·7-s − 0.0844i·8-s + 0.957·9-s + 0.335·11-s + 0.219i·12-s + 0.277i·13-s − 0.229·14-s − 0.937·16-s − 1.84i·17-s + 1.37i·18-s + 1.25·19-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

\(L(3)\) \(\approx\) \(2.252141960\)
\(L(\frac12)\) \(\approx\) \(2.252141960\)
\(L(\frac{7}{2})\) 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 \( 1 - 8.11iT - 32T^{2} \)
3 \( 1 + 3.22iT - 243T^{2} \)
7 \( 1 - 20.7iT - 1.68e4T^{2} \)
11 \( 1 - 134.T + 1.61e5T^{2} \)
17 \( 1 + 2.19e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.98e3T + 2.47e6T^{2} \)
23 \( 1 + 4.22e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.54e3T + 2.05e7T^{2} \)
31 \( 1 + 1.36e3T + 2.86e7T^{2} \)
37 \( 1 + 1.41e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.17e4T + 1.15e8T^{2} \)
43 \( 1 - 6.62e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.48e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.42e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.43e3T + 7.14e8T^{2} \)
61 \( 1 - 1.62e4T + 8.44e8T^{2} \)
67 \( 1 + 1.67e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.71e4T + 1.80e9T^{2} \)
73 \( 1 - 6.32e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.81e4T + 3.07e9T^{2} \)
83 \( 1 + 1.21e5iT - 3.93e9T^{2} \)
89 \( 1 - 4.98e4T + 5.58e9T^{2} \)
97 \( 1 - 2.13e4iT - 8.58e9T^{2} \)
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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

−10.95502841591742620273268747690, −9.545075841507111889019881929559, −8.994183458009201841098726822970, −7.60479194631040237126053437342, −7.23328775008620077425614016519, −6.26043740264983442479912369523, −5.19288597930751357174712064821, −4.26138279276581883662092776084, −2.47038931253312622074028517608, −0.72583150274931120706694165845, 1.05602310791307346173127558315, 1.85393009214672584364261957931, 3.44799612437523565174066540821, 3.99137673253118656230178202746, 5.35931966135605886508121578247, 6.78783614182216681316962834359, 7.897020832011317533456648726660, 9.235300872652246705587873852888, 9.923672135946090102210867496329, 10.60109323713698264517285800830

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