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Properties

Label 2-65-13.4-c3-0-1
Degree $2$
Conductor $65$
Sign $0.606 - 0.795i$
Analytic cond. $3.83512$
Root an. cond. $1.95834$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.63 − 1.52i)2-s + (−1.55 + 2.68i)3-s + (0.622 + 1.07i)4-s − 5i·5-s + (8.16 − 4.71i)6-s + (−6.56 + 3.79i)7-s + 20.5i·8-s + (8.69 + 15.0i)9-s + (−7.60 + 13.1i)10-s + (27.6 + 15.9i)11-s − 3.86·12-s + (17.5 + 43.4i)13-s + 23.0·14-s + (13.4 + 7.75i)15-s + (36.2 − 62.7i)16-s + (18.7 + 32.4i)17-s + ⋯
L(s)  = 1  + (−0.931 − 0.537i)2-s + (−0.298 + 0.516i)3-s + (0.0778 + 0.134i)4-s − 0.447i·5-s + (0.555 − 0.320i)6-s + (−0.354 + 0.204i)7-s + 0.907i·8-s + (0.322 + 0.557i)9-s + (−0.240 + 0.416i)10-s + (0.758 + 0.437i)11-s − 0.0928·12-s + (0.373 + 0.927i)13-s + 0.440·14-s + (0.231 + 0.133i)15-s + (0.565 − 0.979i)16-s + (0.267 + 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.606 - 0.795i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ 0.606 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.588124 + 0.291123i\)
\(L(\frac12)\) \(\approx\) \(0.588124 + 0.291123i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5iT \)
13 \( 1 + (-17.5 - 43.4i)T \)
good2 \( 1 + (2.63 + 1.52i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (1.55 - 2.68i)T + (-13.5 - 23.3i)T^{2} \)
7 \( 1 + (6.56 - 3.79i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-27.6 - 15.9i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-18.7 - 32.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-6.20 + 3.58i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (86.2 - 149. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (48.2 - 83.5i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 280. iT - 2.97e4T^{2} \)
37 \( 1 + (-10.7 - 6.22i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-397. - 229. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (91.4 + 158. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 234. iT - 1.03e5T^{2} \)
53 \( 1 - 28.8T + 1.48e5T^{2} \)
59 \( 1 + (205. - 118. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-95.8 - 165. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-598. - 345. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (576. - 332. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 795. iT - 3.89e5T^{2} \)
79 \( 1 - 232.T + 4.93e5T^{2} \)
83 \( 1 + 889. iT - 5.71e5T^{2} \)
89 \( 1 + (-970. - 560. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-679. + 392. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.61839114246603472134792217856, −13.36047692968075134861464617536, −11.89663885676255047836375826091, −11.02459633218201953670610301371, −9.750754715398359789884100184965, −9.237123257438521255762199638255, −7.76998746632037041175497314368, −5.81702520392081713185238576000, −4.26846872645017515145071006175, −1.68292480434923828067019136813, 0.67407410878215274185516902387, 3.63098863006101186891140750193, 6.19234839899541501052519412370, 7.01794232703356953831141719911, 8.227770725674600721326404455563, 9.466623407463806314998958524508, 10.54934635027113173983658852309, 12.07706421230715027667063163608, 13.00083745570641060455980320575, 14.33191789351496736272086785874

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