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Properties

Label 2-325-65.63-c1-0-14
Degree $2$
Conductor $325$
Sign $0.600 + 0.799i$
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  + (0.403 + 0.698i)2-s + (0.724 − 2.70i)3-s + (0.674 − 1.16i)4-s + (2.17 − 0.584i)6-s + (3.71 + 2.14i)7-s + 2.70·8-s + (−4.18 − 2.41i)9-s + (−3.05 − 0.818i)11-s + (−2.67 − 2.67i)12-s + (−1.10 + 3.43i)13-s + 3.45i·14-s + (−0.260 − 0.450i)16-s + (−4.86 + 1.30i)17-s − 3.89i·18-s + (0.287 + 1.07i)19-s + ⋯
L(s)  = 1  + (0.285 + 0.493i)2-s + (0.418 − 1.56i)3-s + (0.337 − 0.584i)4-s + (0.889 − 0.238i)6-s + (1.40 + 0.809i)7-s + 0.955·8-s + (−1.39 − 0.805i)9-s + (−0.920 − 0.246i)11-s + (−0.770 − 0.770i)12-s + (−0.306 + 0.951i)13-s + 0.923i·14-s + (−0.0650 − 0.112i)16-s + (−1.18 + 0.316i)17-s − 0.918i·18-s + (0.0658 + 0.245i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79281 - 0.895859i\)
\(L(\frac12)\) \(\approx\) \(1.79281 - 0.895859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 - 3.43i)T \)
good2 \( 1 + (-0.403 - 0.698i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.724 + 2.70i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.71 - 2.14i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.05 + 0.818i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (4.86 - 1.30i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.287 - 1.07i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.05 - 1.08i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.02 - 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.69 + 3.69i)T + 31iT^{2} \)
37 \( 1 + (-0.626 + 0.361i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.382 + 1.42i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.34 - 8.76i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 1.60iT - 47T^{2} \)
53 \( 1 + (9.02 + 9.02i)T + 53iT^{2} \)
59 \( 1 + (-0.00962 + 0.00257i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.39 - 4.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.50 - 2.60i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.96 + 0.525i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 14.3T + 73T^{2} \)
79 \( 1 + 2.02iT - 79T^{2} \)
83 \( 1 + 0.914iT - 83T^{2} \)
89 \( 1 + (3.20 - 11.9i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.84 - 4.93i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.36903378758113346663654169330, −11.06287171675801722675327103390, −9.280733178707668655925563819011, −8.259746069831840258973643125358, −7.57375284759195836509199642592, −6.71436327087699277075240646980, −5.72302211680346240651225520952, −4.76673556541286531647335512403, −2.35622375493896274073418628253, −1.65190118834309398270795326649, 2.36816712928188416112157449337, 3.53713445913972194916895992820, 4.60787638306704016789411777647, 5.07673468260368592015967329132, 7.29672960887857359179503690841, 8.031578761916790487786705142363, 8.998787770963768566561252874655, 10.27874112303757131619659760142, 10.87579903489492394930320727589, 11.24783720327779738584532384334

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