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Properties

Label 2-65-13.9-c3-0-11
Degree $2$
Conductor $65$
Sign $-0.798 + 0.602i$
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  + (2.26 − 3.91i)2-s + (0.652 − 1.13i)3-s + (−6.23 − 10.7i)4-s − 5·5-s + (−2.95 − 5.11i)6-s + (−11.9 − 20.7i)7-s − 20.2·8-s + (12.6 + 21.9i)9-s + (−11.3 + 19.5i)10-s + (29.3 − 50.7i)11-s − 16.2·12-s + (17.8 + 43.3i)13-s − 108.·14-s + (−3.26 + 5.65i)15-s + (4.14 − 7.17i)16-s + (60.0 + 104. i)17-s + ⋯
L(s)  = 1  + (0.799 − 1.38i)2-s + (0.125 − 0.217i)3-s + (−0.779 − 1.34i)4-s − 0.447·5-s + (−0.200 − 0.347i)6-s + (−0.647 − 1.12i)7-s − 0.893·8-s + (0.468 + 0.811i)9-s + (−0.357 + 0.619i)10-s + (0.803 − 1.39i)11-s − 0.391·12-s + (0.380 + 0.924i)13-s − 2.07·14-s + (−0.0561 + 0.0972i)15-s + (0.0647 − 0.112i)16-s + (0.856 + 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.798 + 0.602i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ -0.798 + 0.602i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.635544 - 1.89858i\)
\(L(\frac12)\) \(\approx\) \(0.635544 - 1.89858i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
13 \( 1 + (-17.8 - 43.3i)T \)
good2 \( 1 + (-2.26 + 3.91i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-0.652 + 1.13i)T + (-13.5 - 23.3i)T^{2} \)
7 \( 1 + (11.9 + 20.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-29.3 + 50.7i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-60.0 - 104. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (27.3 + 47.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (36.1 - 62.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-85.9 + 148. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 33.9T + 2.97e4T^{2} \)
37 \( 1 + (15.0 - 26.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (168. - 291. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-187. - 325. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 521.T + 1.03e5T^{2} \)
53 \( 1 + 178.T + 1.48e5T^{2} \)
59 \( 1 + (94.7 + 164. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-67.5 - 116. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (174. - 301. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-128. - 223. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 754.T + 3.89e5T^{2} \)
79 \( 1 - 22.0T + 4.93e5T^{2} \)
83 \( 1 + 519.T + 5.71e5T^{2} \)
89 \( 1 + (-451. + 781. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-712. - 1.23e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−13.59336654545178398801770267069, −12.97335005428256900701116843152, −11.64229698544764258358857124231, −10.86246928200777803492762655552, −9.848224476014564751982252601980, −8.084524843791728127413615845034, −6.38888150724871705424886044114, −4.31929722755049925483568660235, −3.42679655107566852901301940178, −1.28453646727851740408153352849, 3.52363065980941833191124204333, 4.97889600232910710589811936686, 6.32918124030600407357699222519, 7.31909371063226668695512426100, 8.757886889923972887052935409015, 9.930720220922073093157577131570, 12.29610749943570172241016283578, 12.47104913698910340289741353972, 14.17016754176341351047793112674, 15.06291056935854090616378812474

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