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Properties

Label 2-325-65.58-c1-0-0
Degree $2$
Conductor $325$
Sign $-0.0453 - 0.998i$
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  + (−1.37 − 0.792i)2-s + (−0.190 + 0.0510i)3-s + (0.255 + 0.442i)4-s + (0.302 + 0.0809i)6-s + (−0.274 − 0.474i)7-s + 2.35i·8-s + (−2.56 + 1.48i)9-s + (0.147 − 0.0396i)11-s + (−0.0713 − 0.0713i)12-s + (−3.21 − 1.63i)13-s + 0.868i·14-s + (2.38 − 4.12i)16-s + (−0.813 + 3.03i)17-s + 4.69·18-s + (−1.18 + 4.40i)19-s + ⋯
L(s)  = 1  + (−0.970 − 0.560i)2-s + (−0.110 + 0.0294i)3-s + (0.127 + 0.221i)4-s + (0.123 + 0.0330i)6-s + (−0.103 − 0.179i)7-s + 0.834i·8-s + (−0.854 + 0.493i)9-s + (0.0446 − 0.0119i)11-s + (−0.0205 − 0.0205i)12-s + (−0.891 − 0.452i)13-s + 0.232i·14-s + (0.595 − 1.03i)16-s + (−0.197 + 0.736i)17-s + 1.10·18-s + (−0.270 + 1.01i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0453 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.0453 - 0.998i$
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: \((2,\ 325,\ (\ :1/2),\ -0.0453 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186759 + 0.195432i\)
\(L(\frac12)\) \(\approx\) \(0.186759 + 0.195432i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.21 + 1.63i)T \)
good2 \( 1 + (1.37 + 0.792i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.190 - 0.0510i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.274 + 0.474i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.147 + 0.0396i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.813 - 3.03i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.18 - 4.40i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.916 - 3.41i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.02 - 1.17i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.60 - 6.60i)T - 31iT^{2} \)
37 \( 1 + (3.40 - 5.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.926 - 3.45i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.86 + 1.84i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.13T + 47T^{2} \)
53 \( 1 + (-3.70 - 3.70i)T + 53iT^{2} \)
59 \( 1 + (3.67 + 0.985i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.92 + 6.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.23 + 2.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (15.1 + 4.04i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 3.91iT - 73T^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + (-2.35 - 8.78i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.55 - 3.78i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.64681162550371799314437645634, −10.56721932187078984587183077104, −10.25787832957628711791722056707, −9.056871204366811675056959611041, −8.332470922509518696290470739289, −7.40279981195160556404218601294, −5.88233593269630485458455834547, −4.95585674404231745221950561516, −3.18025942270245824366910799031, −1.76762421682442138208942422974, 0.25819920228262818726954163330, 2.66960895629840017930008052447, 4.26287267679813242674224483860, 5.69787029569031909269088543802, 6.83444386671984767335777237797, 7.49289133436671544964874225842, 8.868070089939220188238737452462, 9.078187700686431725366415531606, 10.18039543650851020874002600661, 11.31305458079343193130970078570

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