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Properties

Label 2-325-65.58-c1-0-3
Degree $2$
Conductor $325$
Sign $0.916 - 0.400i$
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  + (−2.40 − 1.39i)2-s + (0.732 − 0.196i)3-s + (2.86 + 4.96i)4-s + (−2.03 − 0.545i)6-s + (0.989 + 1.71i)7-s − 10.3i·8-s + (−2.10 + 1.21i)9-s + (2.64 − 0.707i)11-s + (3.07 + 3.07i)12-s + (−2.22 + 2.83i)13-s − 5.50i·14-s + (−8.71 + 15.0i)16-s + (−1.09 + 4.08i)17-s + 6.74·18-s + (−0.742 + 2.77i)19-s + ⋯
L(s)  = 1  + (−1.70 − 0.983i)2-s + (0.422 − 0.113i)3-s + (1.43 + 2.48i)4-s + (−0.831 − 0.222i)6-s + (0.373 + 0.647i)7-s − 3.67i·8-s + (−0.700 + 0.404i)9-s + (0.796 − 0.213i)11-s + (0.887 + 0.887i)12-s + (−0.618 + 0.785i)13-s − 1.47i·14-s + (−2.17 + 3.77i)16-s + (−0.265 + 0.990i)17-s + 1.58·18-s + (−0.170 + 0.635i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594945 + 0.124476i\)
\(L(\frac12)\) \(\approx\) \(0.594945 + 0.124476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 2.83i)T \)
good2 \( 1 + (2.40 + 1.39i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.732 + 0.196i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.989 - 1.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.64 + 0.707i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.09 - 4.08i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.742 - 2.77i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.847 - 3.16i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.288 - 0.166i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.77 + 3.77i)T - 31iT^{2} \)
37 \( 1 + (-0.311 + 0.539i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.894 - 3.33i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.21 - 2.20i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 5.83T + 47T^{2} \)
53 \( 1 + (3.61 + 3.61i)T + 53iT^{2} \)
59 \( 1 + (7.72 + 2.07i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.34 - 9.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.989 + 0.571i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.43 + 1.72i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 4.34iT - 73T^{2} \)
79 \( 1 - 4.93iT - 79T^{2} \)
83 \( 1 - 2.83T + 83T^{2} \)
89 \( 1 + (1.09 + 4.07i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.06 + 5.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.54059868148415102966201097558, −10.71464597291855206379529850089, −9.619358118528718850202368391813, −8.922589510914158692256207335716, −8.283290760627646066878766839411, −7.43820401238795152493203106143, −6.15920743117702652790001412998, −3.99340004946958061129253089232, −2.64628615553345176519771624518, −1.68493232645163641704289376954, 0.72536732297640508592625551859, 2.58096298308969495190405929070, 4.82718382926233899460624027077, 6.11105901229674715121371181410, 7.08295624909933827580847613326, 7.79254717518643503677845386754, 8.830367910778784644325070977775, 9.313285399221144442108975743255, 10.33214894898547214363341507153, 11.08312837055001339999677312908

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