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Properties

Label 2-325-13.4-c1-0-18
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: $\chi_{325} (251, \cdot )$
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

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

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