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Properties

Label 2-325-5.4-c5-0-22
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  − 4.30i·2-s + 19.9i·3-s + 13.4·4-s + 85.7·6-s − 125. i·7-s − 195. i·8-s − 153.·9-s − 709.·11-s + 267. i·12-s + 169i·13-s − 541.·14-s − 412.·16-s + 1.92e3i·17-s + 660. i·18-s + 2.16e3·19-s + ⋯
L(s)  = 1  − 0.761i·2-s + 1.27i·3-s + 0.420·4-s + 0.972·6-s − 0.970i·7-s − 1.08i·8-s − 0.631·9-s − 1.76·11-s + 0.537i·12-s + 0.277i·13-s − 0.738·14-s − 0.402·16-s + 1.61i·17-s + 0.480i·18-s + 1.37·19-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

\(L(3)\) \(\approx\) \(1.868762296\)
\(L(\frac12)\) \(\approx\) \(1.868762296\)
\(L(\frac{7}{2})\) 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 \( 1 + 4.30iT - 32T^{2} \)
3 \( 1 - 19.9iT - 243T^{2} \)
7 \( 1 + 125. iT - 1.68e4T^{2} \)
11 \( 1 + 709.T + 1.61e5T^{2} \)
17 \( 1 - 1.92e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.16e3T + 2.47e6T^{2} \)
23 \( 1 + 305. iT - 6.43e6T^{2} \)
29 \( 1 + 4.51e3T + 2.05e7T^{2} \)
31 \( 1 - 4.07e3T + 2.86e7T^{2} \)
37 \( 1 - 1.23e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.44e4T + 1.15e8T^{2} \)
43 \( 1 - 1.71e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.83e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.19e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.98e4T + 7.14e8T^{2} \)
61 \( 1 - 4.49e3T + 8.44e8T^{2} \)
67 \( 1 - 3.04e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.45e4T + 1.80e9T^{2} \)
73 \( 1 - 5.90e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.33e4T + 3.07e9T^{2} \)
83 \( 1 + 8.38e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.24e5T + 5.58e9T^{2} \)
97 \( 1 + 9.38e4iT - 8.58e9T^{2} \)
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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

−10.81758068457538286230131278248, −10.11529049216487537033453336485, −9.717475237158655738298264330070, −8.143820951737986609468676762708, −7.22818184967704845702521039754, −5.81952955395064610627862380348, −4.59649190875203365213043732372, −3.70396176029634519911912063174, −2.74754488050629105512990070761, −1.18182845500242653281802114521, 0.49713505281481723689920794454, 2.19474872970745452066655912958, 2.76928972504518523842207236535, 5.36993818307635333805859072943, 5.60707924253176589405885651887, 7.05496998049196721546709166870, 7.50073226854556392182206880184, 8.238539906101703305680510650233, 9.447609336972541924403123672402, 10.80989694327476762588141881127

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