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Properties

Label 2-325-5.4-c5-0-0
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  − 10.1i·2-s − 23.0i·3-s − 71.4·4-s − 234.·6-s − 51.0i·7-s + 401. i·8-s − 288.·9-s − 512.·11-s + 1.64e3i·12-s − 169i·13-s − 519.·14-s + 1.79e3·16-s + 978. i·17-s + 2.93e3i·18-s − 1.35e3·19-s + ⋯
L(s)  = 1  − 1.79i·2-s − 1.47i·3-s − 2.23·4-s − 2.65·6-s − 0.393i·7-s + 2.21i·8-s − 1.18·9-s − 1.27·11-s + 3.30i·12-s − 0.277i·13-s − 0.708·14-s + 1.75·16-s + 0.821i·17-s + 2.13i·18-s − 0.860·19-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

\(L(3)\) \(\approx\) \(0.03565927271\)
\(L(\frac12)\) \(\approx\) \(0.03565927271\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.1iT - 32T^{2} \)
3 \( 1 + 23.0iT - 243T^{2} \)
7 \( 1 + 51.0iT - 1.68e4T^{2} \)
11 \( 1 + 512.T + 1.61e5T^{2} \)
17 \( 1 - 978. iT - 1.41e6T^{2} \)
19 \( 1 + 1.35e3T + 2.47e6T^{2} \)
23 \( 1 + 11.3iT - 6.43e6T^{2} \)
29 \( 1 - 8.05e3T + 2.05e7T^{2} \)
31 \( 1 - 759.T + 2.86e7T^{2} \)
37 \( 1 + 1.27e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.81e4T + 1.15e8T^{2} \)
43 \( 1 + 1.95e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.20e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.38e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.81e4T + 7.14e8T^{2} \)
61 \( 1 - 7.62e3T + 8.44e8T^{2} \)
67 \( 1 - 3.76e3iT - 1.35e9T^{2} \)
71 \( 1 + 7.57e4T + 1.80e9T^{2} \)
73 \( 1 - 8.66e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.15e4T + 3.07e9T^{2} \)
83 \( 1 - 4.68e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.23e3T + 5.58e9T^{2} \)
97 \( 1 - 5.75e4iT - 8.58e9T^{2} \)
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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

−10.57979050213745466927738670276, −10.25287629847047053555629505772, −8.679601692738038288195090799201, −8.056340022975117447383181478706, −6.86694743870784475257741364114, −5.49255324502514397848133366215, −4.14766875618198819825954020424, −2.79419867308098144629615891896, −2.01807941354538640865905245492, −0.929598601928396284561748694266, 0.01155801536580284930525462507, 3.00707448815912806399961138979, 4.55466114018571159554847456152, 4.92543621414545199296707327040, 5.94136962740314497622359387354, 7.00771610518488531955144629018, 8.260437818607678951065681159241, 8.766150450830092125371639424793, 9.849864656474009795496182155992, 10.41867012072670629244876013984

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