Properties

Label 2-6e2-4.3-c32-0-11
Degree $2$
Conductor $36$
Sign $0.716 + 0.697i$
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  + (−6.07e4 − 2.46e4i)2-s + (3.07e9 + 2.99e9i)4-s − 2.49e11·5-s − 4.65e13i·7-s + (−1.12e14 − 2.57e14i)8-s + (1.51e16 + 6.15e15i)10-s − 4.19e16i·11-s − 6.87e17·13-s + (−1.14e18 + 2.82e18i)14-s + (4.88e17 + 1.84e19i)16-s − 4.91e19·17-s − 3.14e20i·19-s + (−7.67e20 − 7.47e20i)20-s + (−1.03e21 + 2.54e21i)22-s + 4.04e21i·23-s + ⋯
L(s)  = 1  + (−0.926 − 0.376i)2-s + (0.716 + 0.697i)4-s − 1.63·5-s − 1.39i·7-s + (−0.400 − 0.916i)8-s + (1.51 + 0.615i)10-s − 0.912i·11-s − 1.03·13-s + (−0.526 + 1.29i)14-s + (0.0264 + 0.999i)16-s − 1.01·17-s − 1.08i·19-s + (−1.17 − 1.14i)20-s + (−0.343 + 0.845i)22-s + 0.659i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.716 + 0.697i$
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: \((2,\ 36,\ (\ :16),\ 0.716 + 0.697i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.1704205001\)
\(L(\frac12)\) \(\approx\) \(0.1704205001\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.07e4 + 2.46e4i)T \)
3 \( 1 \)
good5 \( 1 + 2.49e11T + 2.32e22T^{2} \)
7 \( 1 + 4.65e13iT - 1.10e27T^{2} \)
11 \( 1 + 4.19e16iT - 2.11e33T^{2} \)
13 \( 1 + 6.87e17T + 4.42e35T^{2} \)
17 \( 1 + 4.91e19T + 2.36e39T^{2} \)
19 \( 1 + 3.14e20iT - 8.31e40T^{2} \)
23 \( 1 - 4.04e21iT - 3.76e43T^{2} \)
29 \( 1 + 1.16e23T + 6.26e46T^{2} \)
31 \( 1 - 3.53e23iT - 5.29e47T^{2} \)
37 \( 1 + 7.58e24T + 1.52e50T^{2} \)
41 \( 1 - 2.78e25T + 4.06e51T^{2} \)
43 \( 1 - 1.82e26iT - 1.86e52T^{2} \)
47 \( 1 - 6.77e25iT - 3.21e53T^{2} \)
53 \( 1 + 1.13e27T + 1.50e55T^{2} \)
59 \( 1 + 5.33e27iT - 4.64e56T^{2} \)
61 \( 1 + 5.47e28T + 1.35e57T^{2} \)
67 \( 1 - 6.20e28iT - 2.71e58T^{2} \)
71 \( 1 + 4.47e29iT - 1.73e59T^{2} \)
73 \( 1 - 5.03e29T + 4.22e59T^{2} \)
79 \( 1 - 1.79e30iT - 5.29e60T^{2} \)
83 \( 1 - 2.17e30iT - 2.57e61T^{2} \)
89 \( 1 + 1.19e31T + 2.40e62T^{2} \)
97 \( 1 + 9.02e31T + 3.77e63T^{2} \)
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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

−10.89325511615618064088926875827, −9.426788072724051012540236267464, −8.251511693991273160192453526090, −7.44252712225267197796763746921, −6.79879128177722656773933257286, −4.58334534758067949366438580757, −3.71671014074861849782962198136, −2.81536888126771900561899423752, −1.17954646777589952225050580920, −0.26255831259268091188501795138, 0.15521213874434498792703923759, 1.83357376570859394558121611654, 2.71288816146177766752763448842, 4.26708644863641033020966265524, 5.40594141578227700132232471368, 6.78301880474345058601925974236, 7.68835762253022105126347484944, 8.522863530334324451360761141897, 9.469649926294791967439315096434, 10.78225244043116266389216375901

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