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Properties

Label 2-65-13.10-c3-0-4
Degree $2$
Conductor $65$
Sign $0.997 - 0.0637i$
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  + (−1.24 + 0.720i)2-s + (−1.96 − 3.39i)3-s + (−2.96 + 5.12i)4-s − 5i·5-s + (4.89 + 2.82i)6-s + (30.0 + 17.3i)7-s − 20.0i·8-s + (5.80 − 10.0i)9-s + (3.60 + 6.24i)10-s + (31.9 − 18.4i)11-s + 23.2·12-s + (26.4 + 38.6i)13-s − 50.0·14-s + (−16.9 + 9.80i)15-s + (−9.22 − 15.9i)16-s + (−2.41 + 4.17i)17-s + ⋯
L(s)  = 1  + (−0.441 + 0.254i)2-s + (−0.377 − 0.653i)3-s + (−0.370 + 0.641i)4-s − 0.447i·5-s + (0.333 + 0.192i)6-s + (1.62 + 0.937i)7-s − 0.886i·8-s + (0.214 − 0.372i)9-s + (0.113 + 0.197i)10-s + (0.876 − 0.505i)11-s + 0.558·12-s + (0.564 + 0.825i)13-s − 0.955·14-s + (−0.292 + 0.168i)15-s + (−0.144 − 0.249i)16-s + (−0.0344 + 0.0596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0637i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.997 - 0.0637i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ 0.997 - 0.0637i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.13948 + 0.0363321i\)
\(L(\frac12)\) \(\approx\) \(1.13948 + 0.0363321i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-26.4 - 38.6i)T \)
good2 \( 1 + (1.24 - 0.720i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (1.96 + 3.39i)T + (-13.5 + 23.3i)T^{2} \)
7 \( 1 + (-30.0 - 17.3i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-31.9 + 18.4i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (2.41 - 4.17i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-47.7 - 27.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-24.6 - 42.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (33.4 + 57.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 200. iT - 2.97e4T^{2} \)
37 \( 1 + (68.7 - 39.6i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-26.7 + 15.4i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-34.2 + 59.3i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 235. iT - 1.03e5T^{2} \)
53 \( 1 - 584.T + 1.48e5T^{2} \)
59 \( 1 + (250. + 144. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-150. + 260. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (613. - 353. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (810. + 467. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 903. iT - 3.89e5T^{2} \)
79 \( 1 + 1.28e3T + 4.93e5T^{2} \)
83 \( 1 + 1.46e3iT - 5.71e5T^{2} \)
89 \( 1 + (951. - 549. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (17.1 + 9.90i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−14.34125970184571514616316531457, −13.19261364067801128487627677363, −11.88068379708515703118668954244, −11.60046676570606341789178008277, −9.284834462744311471812758963497, −8.533600111203108531028430285002, −7.42279904327835530937689363446, −5.89578924486909792689789871722, −4.18888083736769773500171674703, −1.36461041866345504593232082424, 1.38226524373078965799236579439, 4.32015120249349089301468834199, 5.33975871701226607881896451027, 7.33702313858895776754634676620, 8.694684753550012098908795257902, 10.19160932097082074202431793083, 10.73622829300904744174290646573, 11.56177150260703687465094269280, 13.60030758881664388282920065594, 14.45739816279067360246485958126

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