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Properties

Label 2-325-13.10-c1-0-3
Degree $2$
Conductor $325$
Sign $0.429 - 0.902i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 0.747i)2-s + (0.0473 + 0.0820i)3-s + (0.118 − 0.204i)4-s + (−0.122 − 0.0708i)6-s + (4.18 + 2.41i)7-s − 2.63i·8-s + (1.49 − 2.59i)9-s + (−0.926 + 0.534i)11-s + 0.0224·12-s + (−0.331 − 3.59i)13-s − 7.21·14-s + (2.20 + 3.82i)16-s + (−1.77 + 3.08i)17-s + 4.47i·18-s + (4.96 + 2.86i)19-s + ⋯
L(s)  = 1  + (−0.915 + 0.528i)2-s + (0.0273 + 0.0473i)3-s + (0.0591 − 0.102i)4-s + (−0.0501 − 0.0289i)6-s + (1.57 + 0.912i)7-s − 0.932i·8-s + (0.498 − 0.863i)9-s + (−0.279 + 0.161i)11-s + 0.00647·12-s + (−0.0918 − 0.995i)13-s − 1.92·14-s + (0.552 + 0.956i)16-s + (−0.431 + 0.747i)17-s + 1.05i·18-s + (1.13 + 0.657i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.429 - 0.902i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.429 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801695 + 0.506202i\)
\(L(\frac12)\) \(\approx\) \(0.801695 + 0.506202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.331 + 3.59i)T \)
good2 \( 1 + (1.29 - 0.747i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0473 - 0.0820i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-4.18 - 2.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.926 - 0.534i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.77 - 3.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.96 - 2.86i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.54 - 6.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.736 + 1.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (-0.0219 + 0.0126i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.232 - 0.133i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.77 - 3.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.51iT - 47T^{2} \)
53 \( 1 + 0.991T + 53T^{2} \)
59 \( 1 + (7.55 + 4.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.48 + 2.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.72 + 3.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 8.78T + 79T^{2} \)
83 \( 1 + 0.725iT - 83T^{2} \)
89 \( 1 + (-11.6 + 6.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.97 - 1.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.78315205321050839048574355788, −10.66169936280390174665222319802, −9.642627156352874896853665140141, −8.855829769501030718630881432760, −7.985742290136780320296617191677, −7.41522061565791225194330220742, −5.99970122417543057419696677040, −4.93090236084010085931784733816, −3.45140504286625278217830712886, −1.44413821416746428084762782485, 1.16004502636301028399679741431, 2.35694104386322082255589904984, 4.56760268297470335747990089097, 5.07483789330370884985756159697, 7.07148781724474581004597313386, 7.79226694092456425388928795225, 8.696401241891194184182972911010, 9.631589559441277263922322379250, 10.74980858615657010696903144071, 11.03963253018002107389627073074

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