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Properties

Label 2-325-65.37-c1-0-12
Degree $2$
Conductor $325$
Sign $0.958 - 0.284i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.958 - 0.284i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.958 - 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49750 + 0.362590i\)
\(L(\frac12)\) \(\approx\) \(2.49750 + 0.362590i\)
\(L(\frac{3}{2})\) 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

−11.87365828906271811301855917774, −10.82902263473165023950992879518, −9.594437526756608603732742717536, −8.728559383895008478513806636547, −8.301295487348698559797042393976, −6.84725606662394394083864973074, −5.33754997862170207875944410616, −4.29465651650452568681850799261, −3.00553280975107337393690568252, −2.65148191372397547409508826359, 1.71271449448353471769079742467, 3.61696231779963715803531807800, 4.06416023648813076989350287342, 5.75960175575807401678370713476, 6.73560066865895250904659082191, 7.61708166082410542414267501489, 8.709033538642588450171008524732, 9.578605817023167896671973005897, 10.41815974438426991991752484074, 11.80797540033269070483803022188

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