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Properties

Label 2-65-13.12-c5-0-9
Degree $2$
Conductor $65$
Sign $0.0811 + 0.996i$
Analytic cond. $10.4249$
Root an. cond. $3.22876$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.56i·2-s − 28.5·3-s + 19.2·4-s + 25i·5-s + 101. i·6-s + 143. i·7-s − 182. i·8-s + 572.·9-s + 89.1·10-s − 392. i·11-s − 550.·12-s + (−607. + 49.4i)13-s + 512.·14-s − 714. i·15-s − 35.9·16-s + 484.·17-s + ⋯
L(s)  = 1  − 0.630i·2-s − 1.83·3-s + 0.602·4-s + 0.447i·5-s + 1.15i·6-s + 1.10i·7-s − 1.01i·8-s + 2.35·9-s + 0.282·10-s − 0.976i·11-s − 1.10·12-s + (−0.996 + 0.0811i)13-s + 0.698·14-s − 0.819i·15-s − 0.0350·16-s + 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0811 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0811 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.0811 + 0.996i$
Analytic conductor: \(10.4249\)
Root analytic conductor: \(3.22876\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :5/2),\ 0.0811 + 0.996i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.729447 - 0.672485i\)
\(L(\frac12)\) \(\approx\) \(0.729447 - 0.672485i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 25iT \)
13 \( 1 + (607. - 49.4i)T \)
good2 \( 1 + 3.56iT - 32T^{2} \)
3 \( 1 + 28.5T + 243T^{2} \)
7 \( 1 - 143. iT - 1.68e4T^{2} \)
11 \( 1 + 392. iT - 1.61e5T^{2} \)
17 \( 1 - 484.T + 1.41e6T^{2} \)
19 \( 1 + 3.07e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.91e3T + 6.43e6T^{2} \)
29 \( 1 - 4.31e3T + 2.05e7T^{2} \)
31 \( 1 + 2.95e3iT - 2.86e7T^{2} \)
37 \( 1 + 6.95e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.63e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.56e4T + 1.47e8T^{2} \)
47 \( 1 + 1.52e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.04e4T + 4.18e8T^{2} \)
59 \( 1 + 2.08e4iT - 7.14e8T^{2} \)
61 \( 1 + 650.T + 8.44e8T^{2} \)
67 \( 1 + 9.52e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.53e4iT - 1.80e9T^{2} \)
73 \( 1 + 4.98e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.91e3T + 3.07e9T^{2} \)
83 \( 1 + 1.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.27e4iT - 5.58e9T^{2} \)
97 \( 1 + 3.30e4iT - 8.58e9T^{2} \)
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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

−13.02307395810763118012928148724, −12.09894845688506541245615179857, −11.34415961622251389746989072317, −10.78771462007950654172274366113, −9.458327033090688109435339577564, −7.12226933031964304339881483467, −6.18433517971415813905193347345, −5.00935273626032846830103082247, −2.70054203218576788367006441992, −0.66146591467384761655772876858, 1.23280205251044607049716044386, 4.56143890372930453828251274180, 5.60285353023722427607784776410, 6.87597668768682448447969254480, 7.57721839869209859025319638532, 10.04720602904560664608672188913, 10.73893237542796609682795234832, 12.04625282686386073219512661077, 12.57255773972694059115438343119, 14.36965014530062324846346183411

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