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Properties

Label 2-325-13.9-c1-0-15
Degree $2$
Conductor $325$
Sign $0.564 + 0.825i$
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  + (−0.165 + 0.286i)2-s + (1.34 − 2.33i)3-s + (0.945 + 1.63i)4-s + (0.445 + 0.771i)6-s + (−1.67 − 2.90i)7-s − 1.28·8-s + (−2.12 − 3.67i)9-s + (1.62 − 2.81i)11-s + 5.08·12-s + (3.39 − 1.21i)13-s + 1.10·14-s + (−1.67 + 2.90i)16-s + (0.974 + 1.68i)17-s + 1.40·18-s + (0.622 + 1.07i)19-s + ⋯
L(s)  = 1  + (−0.116 + 0.202i)2-s + (0.777 − 1.34i)3-s + (0.472 + 0.818i)4-s + (0.181 + 0.314i)6-s + (−0.633 − 1.09i)7-s − 0.455·8-s + (−0.707 − 1.22i)9-s + (0.489 − 0.847i)11-s + 1.46·12-s + (0.941 − 0.337i)13-s + 0.296·14-s + (−0.419 + 0.726i)16-s + (0.236 + 0.409i)17-s + 0.331·18-s + (0.142 + 0.247i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.564 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46655 - 0.774033i\)
\(L(\frac12)\) \(\approx\) \(1.46655 - 0.774033i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.39 + 1.21i)T \)
good2 \( 1 + (0.165 - 0.286i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.34 + 2.33i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.67 + 2.90i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.62 + 2.81i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.974 - 1.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.622 - 1.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.34 - 2.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.78T + 31T^{2} \)
37 \( 1 + (0.974 - 1.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.39 - 2.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.36 - 7.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.86T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + (-1.26 - 2.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.74 - 6.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.00 + 3.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.62 + 4.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.46T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 8.61T + 83T^{2} \)
89 \( 1 + (5.15 - 8.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.63 - 4.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.64907592748408028254663983591, −10.68762201553688330072473307007, −9.272616641568434135134267704555, −8.193643156552614233279949409168, −7.79776592445192325834832770320, −6.71509159984012959426896318795, −6.21506791143344672672721428024, −3.73443216338418249785763348930, −3.06054334795429395109780906596, −1.31237083377814264592273014348, 2.18553822503122477012390471528, 3.32441812126338217459835954609, 4.60965111359895253963818418636, 5.74945805622263169193966687269, 6.75626455983017038889950930936, 8.432363825169864539162751742253, 9.365571050989130734931998969586, 9.643579598459333194349844235836, 10.60719732064113102229655767352, 11.54648978569607864815557799725

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