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Properties

Label 2-65-13.3-c1-0-3
Degree $2$
Conductor $65$
Sign $-0.252 + 0.967i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.651 − 1.12i)2-s + (−0.5 − 0.866i)3-s + (0.151 − 0.262i)4-s − 5-s + (−0.651 + 1.12i)6-s + (0.5 − 0.866i)7-s − 3·8-s + (1 − 1.73i)9-s + (0.651 + 1.12i)10-s + (2.80 + 4.85i)11-s − 0.302·12-s + 3.60·13-s − 1.30·14-s + (0.5 + 0.866i)15-s + (1.65 + 2.86i)16-s + (−0.197 + 0.341i)17-s + ⋯
L(s)  = 1  + (−0.460 − 0.797i)2-s + (−0.288 − 0.499i)3-s + (0.0756 − 0.131i)4-s − 0.447·5-s + (−0.265 + 0.460i)6-s + (0.188 − 0.327i)7-s − 1.06·8-s + (0.333 − 0.577i)9-s + (0.205 + 0.356i)10-s + (0.845 + 1.46i)11-s − 0.0874·12-s + 1.00·13-s − 0.348·14-s + (0.129 + 0.223i)15-s + (0.412 + 0.715i)16-s + (−0.0478 + 0.0828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420340 - 0.544176i\)
\(L(\frac12)\) \(\approx\) \(0.420340 - 0.544176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
13 \( 1 - 3.60T \)
good2 \( 1 + (0.651 + 1.12i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.80 - 4.85i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.197 - 0.341i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.802 - 1.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.10 + 7.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.80 - 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.10 - 3.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.21T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (5.40 - 9.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.40 + 14.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 9.21T + 79T^{2} \)
83 \( 1 - 5.21T + 83T^{2} \)
89 \( 1 + (4.10 + 7.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.80 + 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.77897557482352577259779981659, −13.10589483965291201465374908671, −12.04116950388572674951618047530, −11.38273753470751407937937588343, −10.09638830681397758136885600406, −9.078392531186914942929991916857, −7.36207989365010637180779909424, −6.18456918179466007260633386603, −3.95016738525998569579110113141, −1.51545986592200855846070605594, 3.63455688525051791274289816000, 5.57902814091702612338913124920, 6.87919402287606897317926620573, 8.308002667946383299678719688113, 9.036532808057901851766936685973, 10.87905492944601137255752967700, 11.58835307892041387538734409139, 13.06925542886101611705099764002, 14.49399711499434791262040728068, 15.57877162505942249857984663148

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