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Properties

Label 2-6e2-4.3-c32-0-44
Degree $2$
Conductor $36$
Sign $-0.0106 + 0.999i$
Analytic cond. $233.519$
Root an. cond. $15.2813$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.60e4 − 4.65e4i)2-s + (−4.59e7 + 4.29e9i)4-s + 3.25e10·5-s − 4.74e13i·7-s + (2.02e14 − 1.95e14i)8-s + (−1.49e15 − 1.51e15i)10-s + 1.35e16i·11-s + 9.45e16·13-s + (−2.21e18 + 2.18e18i)14-s + (−1.84e19 − 3.94e17i)16-s − 6.35e18·17-s + 4.93e20i·19-s + (−1.49e18 + 1.39e20i)20-s + (6.30e20 − 6.23e20i)22-s − 4.92e21i·23-s + ⋯
L(s)  = 1  + (−0.703 − 0.710i)2-s + (−0.0106 + 0.999i)4-s + 0.213·5-s − 1.42i·7-s + (0.718 − 0.695i)8-s + (−0.149 − 0.151i)10-s + 0.294i·11-s + 0.142·13-s + (−1.01 + 1.00i)14-s + (−0.999 − 0.0213i)16-s − 0.130·17-s + 1.70i·19-s + (−0.00227 + 0.213i)20-s + (0.209 − 0.207i)22-s − 0.803i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.0106 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.0106 + 0.999i$
Analytic conductor: \(233.519\)
Root analytic conductor: \(15.2813\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :16),\ -0.0106 + 0.999i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.467185366\)
\(L(\frac12)\) \(\approx\) \(1.467185366\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.60e4 + 4.65e4i)T \)
3 \( 1 \)
good5 \( 1 - 3.25e10T + 2.32e22T^{2} \)
7 \( 1 + 4.74e13iT - 1.10e27T^{2} \)
11 \( 1 - 1.35e16iT - 2.11e33T^{2} \)
13 \( 1 - 9.45e16T + 4.42e35T^{2} \)
17 \( 1 + 6.35e18T + 2.36e39T^{2} \)
19 \( 1 - 4.93e20iT - 8.31e40T^{2} \)
23 \( 1 + 4.92e21iT - 3.76e43T^{2} \)
29 \( 1 - 2.04e23T + 6.26e46T^{2} \)
31 \( 1 + 8.15e23iT - 5.29e47T^{2} \)
37 \( 1 - 8.64e24T + 1.52e50T^{2} \)
41 \( 1 - 9.37e25T + 4.06e51T^{2} \)
43 \( 1 - 9.13e25iT - 1.86e52T^{2} \)
47 \( 1 + 4.69e26iT - 3.21e53T^{2} \)
53 \( 1 + 3.40e27T + 1.50e55T^{2} \)
59 \( 1 - 2.87e27iT - 4.64e56T^{2} \)
61 \( 1 - 7.72e27T + 1.35e57T^{2} \)
67 \( 1 - 8.28e28iT - 2.71e58T^{2} \)
71 \( 1 - 2.37e29iT - 1.73e59T^{2} \)
73 \( 1 + 4.61e27T + 4.22e59T^{2} \)
79 \( 1 - 3.06e30iT - 5.29e60T^{2} \)
83 \( 1 - 3.67e29iT - 2.57e61T^{2} \)
89 \( 1 - 2.12e31T + 2.40e62T^{2} \)
97 \( 1 - 2.45e30T + 3.77e63T^{2} \)
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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

−10.28538909162595059894743646132, −9.677387974997778892781390462360, −8.190444729896672573818673709122, −7.45612412036715970470354386585, −6.22433535673918423535798500675, −4.37590035722465997073699966388, −3.72897197815803470026649709145, −2.37724068238540346312890495362, −1.31694612261224416430080193698, −0.49627227092805262359125636938, 0.67039797446511935074594647516, 1.89535365227887982040544461347, 2.85502757245230001428902413270, 4.74282675145372499052034824479, 5.68233585601631422770804868452, 6.50761925819990625370595445284, 7.78335508452793511979962195398, 8.902141680706333767860467569692, 9.415869366264155457780102337915, 10.83452747018938420350470737510

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