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Properties

Label 2-65-13.6-c2-0-1
Degree $2$
Conductor $65$
Sign $-0.128 - 0.991i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.107 + 0.401i)2-s + (−2.08 + 3.61i)3-s + (3.31 − 1.91i)4-s + (−1.58 + 1.58i)5-s + (−1.67 − 0.448i)6-s + (−2.68 + 10.0i)7-s + (2.30 + 2.30i)8-s + (−4.19 − 7.26i)9-s + (−0.804 − 0.464i)10-s + (−4.73 + 1.26i)11-s + 15.9i·12-s + (7.53 − 10.5i)13-s − 4.31·14-s + (−2.41 − 9.00i)15-s + (6.97 − 12.0i)16-s + (19.1 − 11.0i)17-s + ⋯
L(s)  = 1  + (0.0537 + 0.200i)2-s + (−0.695 + 1.20i)3-s + (0.828 − 0.478i)4-s + (−0.316 + 0.316i)5-s + (−0.279 − 0.0747i)6-s + (−0.384 + 1.43i)7-s + (0.287 + 0.287i)8-s + (−0.466 − 0.807i)9-s + (−0.0804 − 0.0464i)10-s + (−0.430 + 0.115i)11-s + 1.33i·12-s + (0.579 − 0.814i)13-s − 0.308·14-s + (−0.160 − 0.600i)15-s + (0.436 − 0.755i)16-s + (1.12 − 0.650i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.128 - 0.991i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ -0.128 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.730011 + 0.830567i\)
\(L(\frac12)\) \(\approx\) \(0.730011 + 0.830567i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.58 - 1.58i)T \)
13 \( 1 + (-7.53 + 10.5i)T \)
good2 \( 1 + (-0.107 - 0.401i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (2.08 - 3.61i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + (2.68 - 10.0i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (4.73 - 1.26i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-19.1 + 11.0i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-33.5 - 8.98i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (27.6 + 15.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (13.5 - 23.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-9.08 + 9.08i)T - 961iT^{2} \)
37 \( 1 + (-14.3 + 3.85i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (3.69 + 13.7i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (5.13 - 2.96i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-33.5 - 33.5i)T + 2.20e3iT^{2} \)
53 \( 1 + 33.8T + 2.80e3T^{2} \)
59 \( 1 + (-9.82 + 36.6i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (9.55 + 16.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-11.6 - 43.6i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (63.4 + 17.0i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (55.4 + 55.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 71.0T + 6.24e3T^{2} \)
83 \( 1 + (54.0 - 54.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (66.8 - 17.9i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-55.3 - 14.8i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−15.36165848420685372915759940607, −14.32083783001265490200604294869, −12.26759695429647581716801989590, −11.50276289277548162460888908287, −10.40072127724824896999107397752, −9.591311676652727353042912228984, −7.76427627498086041655175404827, −5.91247800154617768946078233362, −5.33464943756115517646949841931, −3.08870635636619729732040670380, 1.23927448575699672440405269192, 3.69959741108024439808372380306, 6.02441818996519904417717533179, 7.25032594696417270434962362220, 7.79322343680294846130589736608, 10.08186597543692861907462934190, 11.39692042424611468569555805579, 11.99909123349522542361731540269, 13.13249565778300918262227107514, 13.82403232368570570477117071089

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