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Properties

Label 2-325-13.10-c1-0-0
Degree $2$
Conductor $325$
Sign $-0.875 + 0.484i$
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  + (−1.05 + 0.609i)2-s + (1.16 + 2.01i)3-s + (−0.256 + 0.443i)4-s + (−2.46 − 1.42i)6-s + (−3.11 − 1.80i)7-s − 3.06i·8-s + (−1.21 + 2.11i)9-s + (−4.65 + 2.68i)11-s − 1.19·12-s + (−1.81 + 3.11i)13-s + 4.39·14-s + (1.35 + 2.34i)16-s + (0.565 − 0.980i)17-s − 2.97i·18-s + (−1.96 − 1.13i)19-s + ⋯
L(s)  = 1  + (−0.746 + 0.431i)2-s + (0.673 + 1.16i)3-s + (−0.128 + 0.221i)4-s + (−1.00 − 0.580i)6-s + (−1.17 − 0.680i)7-s − 1.08i·8-s + (−0.406 + 0.704i)9-s + (−1.40 + 0.809i)11-s − 0.344·12-s + (−0.504 + 0.863i)13-s + 1.17·14-s + (0.339 + 0.587i)16-s + (0.137 − 0.237i)17-s − 0.701i·18-s + (−0.450 − 0.260i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.875 + 0.484i$
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: \((2,\ 325,\ (\ :1/2),\ -0.875 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116185 - 0.450051i\)
\(L(\frac12)\) \(\approx\) \(0.116185 - 0.450051i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 - 3.11i)T \)
good2 \( 1 + (1.05 - 0.609i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.16 - 2.01i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (3.11 + 1.80i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.65 - 2.68i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.565 + 0.980i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.94 - 3.37i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0123 - 0.0214i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + (7.53 - 4.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.23 + 1.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.565 + 0.980i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.58iT - 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + (0.148 + 0.0857i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 - 2.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.54 - 3.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.35 - 5.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.70iT - 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 + (-13.9 + 8.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.5 - 6.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.28057104325395985198122133891, −10.57035383157930320892821323831, −10.00587410604729364836799541269, −9.416865834877204505818391936325, −8.631528886242616695522641277171, −7.44299485486670064218788533857, −6.80105973127391386506589999838, −4.92952415145728133452906305196, −3.92589180145384844113079018261, −2.91763605169742869506362369386, 0.36699163455488236509834523873, 2.27248563884518143978319131404, 2.99952351845381476116035899568, 5.34287827799795065241538381853, 6.23453097315273726050863575399, 7.60029422662324823162133010468, 8.316053609340632777374437815363, 9.058911798118225173289105897448, 10.13656082798564826131406422330, 10.78897785309335828626809491425

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