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Properties

Label 2-65-13.4-c3-0-9
Degree $2$
Conductor $65$
Sign $-0.589 + 0.807i$
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  + (0.172 + 0.0994i)2-s + (−4.74 + 8.21i)3-s + (−3.98 − 6.89i)4-s − 5i·5-s + (−1.63 + 0.943i)6-s + (0.235 − 0.135i)7-s − 3.17i·8-s + (−31.4 − 54.5i)9-s + (0.497 − 0.861i)10-s + (−27.6 − 15.9i)11-s + 75.4·12-s + (−18.4 − 43.1i)13-s + 0.0540·14-s + (41.0 + 23.7i)15-s + (−31.5 + 54.6i)16-s + (11.2 + 19.4i)17-s + ⋯
L(s)  = 1  + (0.0609 + 0.0351i)2-s + (−0.912 + 1.58i)3-s + (−0.497 − 0.861i)4-s − 0.447i·5-s + (−0.111 + 0.0641i)6-s + (0.0126 − 0.00733i)7-s − 0.140i·8-s + (−1.16 − 2.01i)9-s + (0.0157 − 0.0272i)10-s + (−0.757 − 0.437i)11-s + 1.81·12-s + (−0.392 − 0.919i)13-s + 0.00103·14-s + (0.706 + 0.408i)15-s + (−0.492 + 0.853i)16-s + (0.160 + 0.277i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.589 + 0.807i$
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: \((2,\ 65,\ (\ :3/2),\ -0.589 + 0.807i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0890279 - 0.175230i\)
\(L(\frac12)\) \(\approx\) \(0.0890279 - 0.175230i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.4 + 43.1i)T \)
good2 \( 1 + (-0.172 - 0.0994i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (4.74 - 8.21i)T + (-13.5 - 23.3i)T^{2} \)
7 \( 1 + (-0.235 + 0.135i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (27.6 + 15.9i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-11.2 - 19.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (73.6 - 42.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (80.1 - 138. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-77.1 + 133. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 49.1iT - 2.97e4T^{2} \)
37 \( 1 + (268. + 155. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (71.7 + 41.4i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-160. - 277. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 545. iT - 1.03e5T^{2} \)
53 \( 1 - 114.T + 1.48e5T^{2} \)
59 \( 1 + (-587. + 339. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (188. + 327. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-550. - 317. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (511. - 295. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 1.20e3iT - 3.89e5T^{2} \)
79 \( 1 + 727.T + 4.93e5T^{2} \)
83 \( 1 + 499. iT - 5.71e5T^{2} \)
89 \( 1 + (191. + 110. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-109. + 63.3i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−14.29989772265326500390363756492, −12.85107822916357536581938568614, −11.41359936807531048031453917344, −10.31582273763112869875014699748, −9.844582995552820863274289582728, −8.471021089006415064541000930046, −5.86628887956017933910745790260, −5.24481953182317176494360004493, −3.97691668619161306749161413947, −0.14048720691818988397105512825, 2.37872137380444388825484581423, 4.85574124059610068994216794144, 6.58363369921593923762181003215, 7.40258739176120652321820142933, 8.562063214336128928182898096423, 10.57431696433772604915673870856, 11.86692225027687711624929407923, 12.46779062591851682627854157134, 13.39164904778029208989449750599, 14.31972675861753460212569948021

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