Properties

Label 2-6e2-4.3-c32-0-22
Degree $2$
Conductor $36$
Sign $0.461 - 0.887i$
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  + (5.60e4 − 3.40e4i)2-s + (1.98e9 − 3.81e9i)4-s + 2.17e11·5-s + 2.81e12i·7-s + (−1.85e13 − 2.80e14i)8-s + (1.21e16 − 7.38e15i)10-s + 6.95e16i·11-s − 7.08e17·13-s + (9.58e16 + 1.57e17i)14-s + (−1.05e19 − 1.51e19i)16-s − 6.55e19·17-s − 1.83e20i·19-s + (4.30e20 − 8.27e20i)20-s + (2.36e21 + 3.89e21i)22-s − 7.92e20i·23-s + ⋯
L(s)  = 1  + (0.854 − 0.518i)2-s + (0.461 − 0.887i)4-s + 1.42·5-s + 0.0847i·7-s + (−0.0658 − 0.997i)8-s + (1.21 − 0.738i)10-s + 1.51i·11-s − 1.06·13-s + (0.0440 + 0.0724i)14-s + (−0.574 − 0.818i)16-s − 1.34·17-s − 0.637i·19-s + (0.656 − 1.26i)20-s + (0.785 + 1.29i)22-s − 0.129i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.461 - 0.887i$
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: \((2,\ 36,\ (\ :16),\ 0.461 - 0.887i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(3.093539524\)
\(L(\frac12)\) \(\approx\) \(3.093539524\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.60e4 + 3.40e4i)T \)
3 \( 1 \)
good5 \( 1 - 2.17e11T + 2.32e22T^{2} \)
7 \( 1 - 2.81e12iT - 1.10e27T^{2} \)
11 \( 1 - 6.95e16iT - 2.11e33T^{2} \)
13 \( 1 + 7.08e17T + 4.42e35T^{2} \)
17 \( 1 + 6.55e19T + 2.36e39T^{2} \)
19 \( 1 + 1.83e20iT - 8.31e40T^{2} \)
23 \( 1 + 7.92e20iT - 3.76e43T^{2} \)
29 \( 1 - 1.45e23T + 6.26e46T^{2} \)
31 \( 1 - 1.34e24iT - 5.29e47T^{2} \)
37 \( 1 + 2.62e24T + 1.52e50T^{2} \)
41 \( 1 - 1.80e25T + 4.06e51T^{2} \)
43 \( 1 - 1.56e26iT - 1.86e52T^{2} \)
47 \( 1 - 8.64e26iT - 3.21e53T^{2} \)
53 \( 1 + 5.50e27T + 1.50e55T^{2} \)
59 \( 1 - 9.48e27iT - 4.64e56T^{2} \)
61 \( 1 - 3.04e28T + 1.35e57T^{2} \)
67 \( 1 - 7.40e28iT - 2.71e58T^{2} \)
71 \( 1 + 2.60e29iT - 1.73e59T^{2} \)
73 \( 1 - 3.70e29T + 4.22e59T^{2} \)
79 \( 1 + 2.10e30iT - 5.29e60T^{2} \)
83 \( 1 - 6.24e30iT - 2.57e61T^{2} \)
89 \( 1 + 1.03e31T + 2.40e62T^{2} \)
97 \( 1 - 1.05e32T + 3.77e63T^{2} \)
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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

−10.82886049890279753994236055521, −9.899953866290359074232358554487, −9.209095387062100535055216209599, −7.09128930669927182263081433879, −6.34578732494045037544764641358, −5.03456091062719702060847884063, −4.54266785351539910365784209378, −2.73604280712646512215774529989, −2.17130813241284232934212339650, −1.29043756346559338794551891139, 0.31762525136755202434344394540, 1.96054027110560280212545631220, 2.66317029059693065604099384428, 3.95455776620220748362187180800, 5.23330954193285396517153967204, 5.95100789988003054644663016221, 6.81022226065982468709512718483, 8.197443202524250338627491040069, 9.305210378144218436001475613747, 10.59971371176641366806379951911

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