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Properties

Label 2-65-13.10-c5-0-11
Degree $2$
Conductor $65$
Sign $0.901 - 0.431i$
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  + (−5.67 + 3.27i)2-s + (4.13 + 7.16i)3-s + (5.45 − 9.45i)4-s − 25i·5-s + (−46.9 − 27.1i)6-s + (50.1 + 28.9i)7-s − 138. i·8-s + (87.2 − 151. i)9-s + (81.8 + 141. i)10-s + (111. − 64.5i)11-s + 90.3·12-s + (−507. − 337. i)13-s − 379.·14-s + (179. − 103. i)15-s + (627. + 1.08e3i)16-s + (505. − 876. i)17-s + ⋯
L(s)  = 1  + (−1.00 + 0.579i)2-s + (0.265 + 0.459i)3-s + (0.170 − 0.295i)4-s − 0.447i·5-s + (−0.532 − 0.307i)6-s + (0.386 + 0.223i)7-s − 0.762i·8-s + (0.359 − 0.622i)9-s + (0.258 + 0.448i)10-s + (0.278 − 0.160i)11-s + 0.181·12-s + (−0.832 − 0.554i)13-s − 0.516·14-s + (0.205 − 0.118i)15-s + (0.612 + 1.06i)16-s + (0.424 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.901 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.901 - 0.431i$
Analytic conductor: \(10.4249\)
Root analytic conductor: \(3.22876\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :5/2),\ 0.901 - 0.431i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.10006 + 0.249864i\)
\(L(\frac12)\) \(\approx\) \(1.10006 + 0.249864i\)
\(L(\frac{7}{2})\) 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 \( 1 + (507. + 337. i)T \)
good2 \( 1 + (5.67 - 3.27i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (-4.13 - 7.16i)T + (-121.5 + 210. i)T^{2} \)
7 \( 1 + (-50.1 - 28.9i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-111. + 64.5i)T + (8.05e4 - 1.39e5i)T^{2} \)
17 \( 1 + (-505. + 876. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-2.39e3 - 1.38e3i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.66e3 - 2.89e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-708. - 1.22e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + 6.57e3iT - 2.86e7T^{2} \)
37 \( 1 + (5.02e3 - 2.89e3i)T + (3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + (-1.75e4 + 1.01e4i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (5.57e3 - 9.65e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + 1.58e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.29e4T + 4.18e8T^{2} \)
59 \( 1 + (-2.29e4 - 1.32e4i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.53e4 + 4.38e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.45e4 + 1.41e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-7.01e4 - 4.04e4i)T + (9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 3.72e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.52e4T + 3.07e9T^{2} \)
83 \( 1 + 8.54e3iT - 3.93e9T^{2} \)
89 \( 1 + (-4.28e4 + 2.47e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (1.48e4 + 8.54e3i)T + (4.29e9 + 7.43e9i)T^{2} \)
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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

−14.28662700350932068797067107857, −12.78558490300961006899323447811, −11.69565426126344802406909001769, −9.825953807859363882046127152231, −9.441213095476525639738551657657, −8.146924435344718447316732583054, −7.16594756176145809544016859577, −5.34083893552930917794078516431, −3.57080241346464062023355204288, −0.902692287934127437682677569260, 1.20478014872440545603954627878, 2.57071242634768543004753689040, 4.91748035238130916057186637812, 7.02192766054251673149743994406, 8.007464543463099644316568042965, 9.286105654152413877553661483080, 10.35599514107793147762243050907, 11.26883301352999914602967944670, 12.47608665826438091084688766603, 13.97787837562471313320655928028

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