Properties

Label 2-65-13.9-c3-0-0
Degree $2$
Conductor $65$
Sign $0.419 + 0.907i$
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  + (−2.45 + 4.24i)2-s + (−3.39 + 5.88i)3-s + (−8.03 − 13.9i)4-s − 5·5-s + (−16.6 − 28.8i)6-s + (2.60 + 4.50i)7-s + 39.5·8-s + (−9.56 − 16.5i)9-s + (12.2 − 21.2i)10-s + (−20.9 + 36.3i)11-s + 109.·12-s + (46.1 − 8.28i)13-s − 25.5·14-s + (16.9 − 29.4i)15-s + (−32.7 + 56.6i)16-s + (−19.1 − 33.2i)17-s + ⋯
L(s)  = 1  + (−0.867 + 1.50i)2-s + (−0.653 + 1.13i)3-s + (−1.00 − 1.73i)4-s − 0.447·5-s + (−1.13 − 1.96i)6-s + (0.140 + 0.243i)7-s + 1.74·8-s + (−0.354 − 0.613i)9-s + (0.387 − 0.671i)10-s + (−0.574 + 0.995i)11-s + 2.62·12-s + (0.984 − 0.176i)13-s − 0.487·14-s + (0.292 − 0.506i)15-s + (−0.511 + 0.885i)16-s + (−0.273 − 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.419 + 0.907i$
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: \((2,\ 65,\ (\ :3/2),\ 0.419 + 0.907i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.189364 - 0.121086i\)
\(L(\frac12)\) \(\approx\) \(0.189364 - 0.121086i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-46.1 + 8.28i)T \)
good2 \( 1 + (2.45 - 4.24i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (3.39 - 5.88i)T + (-13.5 - 23.3i)T^{2} \)
7 \( 1 + (-2.60 - 4.50i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (20.9 - 36.3i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (19.1 + 33.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (67.6 + 117. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (73.9 - 128. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-98.4 + 170. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 177.T + 2.97e4T^{2} \)
37 \( 1 + (156. - 271. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (72.0 - 124. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (36.6 + 63.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 194.T + 1.03e5T^{2} \)
53 \( 1 + 751.T + 1.48e5T^{2} \)
59 \( 1 + (-226. - 392. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-77.2 - 133. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (424. - 734. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-379. - 657. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 143.T + 3.89e5T^{2} \)
79 \( 1 + 133.T + 4.93e5T^{2} \)
83 \( 1 + 607.T + 5.71e5T^{2} \)
89 \( 1 + (74.0 - 128. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-247. - 429. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.56692001041415795991193217343, −15.06696534020895449963265649607, −13.44164221096886952047413468219, −11.52952595770689257981995507923, −10.38506527902959359844272981899, −9.417084634137110580945266330213, −8.269173538456002807947940566246, −6.97927941559163008880825552586, −5.57258234022912214914945549714, −4.50758520977939188919479456958, 0.22212635583666826747975457148, 1.69440086574680046972422177571, 3.69779311619164085039415654596, 6.19703231639468474356215263072, 7.894964757674092768909259237410, 8.721331395783892632603013572608, 10.63602954430976018848794660381, 11.02964421389669848965567417992, 12.33848090261875960972855703427, 12.74297435483517370622086971516

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