Properties

Label 2-6e2-12.11-c15-0-27
Degree $2$
Conductor $36$
Sign $-0.577 + 0.816i$
Analytic cond. $51.3696$
Root an. cond. $7.16726$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 181. i·2-s − 3.27e4·4-s − 3.47e5i·5-s − 5.93e6i·8-s + 6.28e7·10-s + 2.57e8·13-s + 1.07e9·16-s − 8.44e8i·17-s + 1.13e10i·20-s − 8.99e10·25-s + 4.65e10i·26-s − 1.81e11i·29-s + 1.94e11i·32-s + 1.52e11·34-s − 7.12e11·37-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 1.98i·5-s − 1.00i·8-s + 1.98·10-s + 1.13·13-s + 1.00·16-s − 0.499i·17-s + 1.98i·20-s − 2.94·25-s + 1.13i·26-s − 1.95i·29-s + 1.00i·32-s + 0.499·34-s − 1.23·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(51.3696\)
Root analytic conductor: \(7.16726\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :15/2),\ -0.577 + 0.816i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.9315064926\)
\(L(\frac12)\) \(\approx\) \(0.9315064926\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 181. iT \)
3 \( 1 \)
good5 \( 1 + 3.47e5iT - 3.05e10T^{2} \)
7 \( 1 - 4.74e12T^{2} \)
11 \( 1 + 4.17e15T^{2} \)
13 \( 1 - 2.57e8T + 5.11e16T^{2} \)
17 \( 1 + 8.44e8iT - 2.86e18T^{2} \)
19 \( 1 - 1.51e19T^{2} \)
23 \( 1 + 2.66e20T^{2} \)
29 \( 1 + 1.81e11iT - 8.62e21T^{2} \)
31 \( 1 - 2.34e22T^{2} \)
37 \( 1 + 7.12e11T + 3.33e23T^{2} \)
41 \( 1 - 2.48e12iT - 1.55e24T^{2} \)
43 \( 1 - 3.17e24T^{2} \)
47 \( 1 + 1.20e25T^{2} \)
53 \( 1 + 1.65e13iT - 7.31e25T^{2} \)
59 \( 1 + 3.65e26T^{2} \)
61 \( 1 + 4.12e13T + 6.02e26T^{2} \)
67 \( 1 - 2.46e27T^{2} \)
71 \( 1 + 5.87e27T^{2} \)
73 \( 1 + 1.17e14T + 8.90e27T^{2} \)
79 \( 1 - 2.91e28T^{2} \)
83 \( 1 + 6.11e28T^{2} \)
89 \( 1 - 8.06e14iT - 1.74e29T^{2} \)
97 \( 1 - 1.56e13T + 6.33e29T^{2} \)
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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

−13.08994901078730415656177658187, −11.86313729231741315333120845859, −9.708070202793863831375290681544, −8.732944897566346424022077318458, −7.930849354132620684539833579049, −6.11110107572196866895088926291, −5.02736576238307663923626686265, −4.00854991431728005237653962229, −1.29941000296180057922452761557, −0.26545807978950465793698805791, 1.65937476535851628954350795831, 2.97228064839521839032095543543, 3.80552064603672791074426702687, 5.86487382495273136690499450309, 7.25497354284728953776955916261, 8.870444910232571093442439011352, 10.55980411974951346611864661819, 10.75440154648173077454318796826, 12.10627065603150604937498279492, 13.64790209305891297871349209755

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