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Properties

Label 2-65-13.9-c1-0-3
Degree $2$
Conductor $65$
Sign $0.252 + 0.967i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
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  + (1.15 − 1.99i)2-s + (−0.5 + 0.866i)3-s + (−1.65 − 2.86i)4-s − 5-s + (1.15 + 1.99i)6-s + (0.5 + 0.866i)7-s − 2.99·8-s + (1 + 1.73i)9-s + (−1.15 + 1.99i)10-s + (−0.802 + 1.39i)11-s + 3.30·12-s − 3.60·13-s + 2.30·14-s + (0.5 − 0.866i)15-s + (−0.151 + 0.262i)16-s + (−3.80 − 6.58i)17-s + ⋯
L(s)  = 1  + (0.814 − 1.41i)2-s + (−0.288 + 0.499i)3-s + (−0.825 − 1.43i)4-s − 0.447·5-s + (0.470 + 0.814i)6-s + (0.188 + 0.327i)7-s − 1.06·8-s + (0.333 + 0.577i)9-s + (−0.364 + 0.630i)10-s + (−0.242 + 0.419i)11-s + 0.953·12-s − 1.00·13-s + 0.615·14-s + (0.129 − 0.223i)15-s + (−0.0378 + 0.0655i)16-s + (−0.922 − 1.59i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.882867 - 0.681955i\)
\(L(\frac12)\) \(\approx\) \(0.882867 - 0.681955i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
13 \( 1 + 3.60T \)
good2 \( 1 + (-1.15 + 1.99i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.802 - 1.39i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.80 + 6.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.80 - 4.85i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.10 + 5.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1.80 - 3.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.10 - 8.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.21T + 47T^{2} \)
53 \( 1 + 3.21T + 53T^{2} \)
59 \( 1 + (-5.40 - 9.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.40 + 4.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.788T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 + 9.21T + 83T^{2} \)
89 \( 1 + (-3.10 + 5.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.19 - 7.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.40805439940454396411539272746, −13.34204889724484478064714586755, −12.19986809291769919145899052411, −11.50564070308548827203924012796, −10.42582258176282826104586338613, −9.520704082235653297044018337427, −7.52540775289160939623334917265, −5.18213621752467731457333050423, −4.36800531468517690847863369810, −2.52759809345947439612794892322, 3.97776970866971482582701838201, 5.39734069479177377929684220852, 6.78905538776817669651575731269, 7.46498094166597590559225299668, 8.850316335612125910152289476152, 10.83936797101497974980434560023, 12.33172832529868215671343923018, 13.09063773268110204174052223411, 14.20033562151667667600496360444, 15.23240186713145171807298469486

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