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Properties

Label 2-325-65.58-c1-0-6
Degree $2$
Conductor $325$
Sign $-0.0868 + 0.996i$
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  + (−1.06 − 0.616i)2-s + (−2.26 + 0.608i)3-s + (−0.240 − 0.416i)4-s + (2.79 + 0.749i)6-s + (1.20 + 2.09i)7-s + 3.05i·8-s + (2.18 − 1.25i)9-s + (0.721 − 0.193i)11-s + (0.798 + 0.798i)12-s + (−1.41 − 3.31i)13-s − 2.98i·14-s + (1.40 − 2.43i)16-s + (1.22 − 4.58i)17-s − 3.10·18-s + (−1.17 + 4.37i)19-s + ⋯
L(s)  = 1  + (−0.754 − 0.435i)2-s + (−1.31 + 0.351i)3-s + (−0.120 − 0.208i)4-s + (1.14 + 0.305i)6-s + (0.456 + 0.791i)7-s + 1.08i·8-s + (0.727 − 0.419i)9-s + (0.217 − 0.0582i)11-s + (0.230 + 0.230i)12-s + (−0.393 − 0.919i)13-s − 0.796i·14-s + (0.350 − 0.607i)16-s + (0.297 − 1.11i)17-s − 0.732·18-s + (−0.269 + 1.00i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0868 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0868 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.0868 + 0.996i$
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: \((2,\ 325,\ (\ :1/2),\ -0.0868 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.276228 - 0.301367i\)
\(L(\frac12)\) \(\approx\) \(0.276228 - 0.301367i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 3.31i)T \)
good2 \( 1 + (1.06 + 0.616i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (2.26 - 0.608i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.20 - 2.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.721 + 0.193i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.22 + 4.58i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.17 - 4.37i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.90 + 7.11i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.31 + 2.49i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.01 + 6.01i)T - 31iT^{2} \)
37 \( 1 + (-0.298 + 0.517i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.40 + 5.25i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-11.4 - 3.05i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + (-0.469 - 0.469i)T + 53iT^{2} \)
59 \( 1 + (-9.49 - 2.54i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.729 + 1.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.37 - 1.94i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.47 + 0.664i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 4.94iT - 73T^{2} \)
79 \( 1 + 10.7iT - 79T^{2} \)
83 \( 1 - 1.96T + 83T^{2} \)
89 \( 1 + (3.40 + 12.6i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-10.1 + 5.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.37602839548828126981950153801, −10.34109856069201714604171453754, −9.871821626072563904754973553319, −8.712008711951690520745065317765, −7.79434161253163997835753170804, −6.10666309219577130295025805973, −5.47895087474666180565301238153, −4.56421372868342784567078120093, −2.39603916106659227892155070204, −0.51104825284433073503675715922, 1.24971658037993543165424737243, 3.87335494363817124596710837175, 4.98325364179310142403713386760, 6.35041518420959986452305409879, 7.03649662705673568559304285986, 7.87297658106885695179651425100, 9.013295125994645005705208873147, 10.01309677384978529330267555215, 10.96786248148063123383945503856, 11.71863723819527621042339121734

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