Properties

Label 2-6e2-4.3-c8-0-16
Degree $2$
Conductor $36$
Sign $-0.218 + 0.975i$
Analytic cond. $14.6656$
Root an. cond. $3.82957$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10 + 12.4i)2-s + (−56 + 249. i)4-s − 610·5-s − 1.39e3i·7-s + (−3.68e3 + 1.79e3i)8-s + (−6.10e3 − 7.61e3i)10-s − 1.84e4i·11-s − 5.47e3·13-s + (1.74e4 − 1.39e4i)14-s + (−5.92e4 − 2.79e4i)16-s − 7.30e4·17-s + 1.94e4i·19-s + (3.41e4 − 1.52e5i)20-s + (2.30e5 − 1.84e5i)22-s − 2.37e5i·23-s + ⋯
L(s)  = 1  + (0.625 + 0.780i)2-s + (−0.218 + 0.975i)4-s − 0.976·5-s − 0.582i·7-s + (−0.898 + 0.439i)8-s + (−0.609 − 0.761i)10-s − 1.26i·11-s − 0.191·13-s + (0.454 − 0.364i)14-s + (−0.904 − 0.426i)16-s − 0.875·17-s + 0.149i·19-s + (0.213 − 0.952i)20-s + (0.985 − 0.789i)22-s − 0.847i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(14.6656\)
Root analytic conductor: \(3.82957\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :4),\ -0.218 + 0.975i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.217791 - 0.272021i\)
\(L(\frac12)\) \(\approx\) \(0.217791 - 0.272021i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-10 - 12.4i)T \)
3 \( 1 \)
good5 \( 1 + 610T + 3.90e5T^{2} \)
7 \( 1 + 1.39e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.84e4iT - 2.14e8T^{2} \)
13 \( 1 + 5.47e3T + 8.15e8T^{2} \)
17 \( 1 + 7.30e4T + 6.97e9T^{2} \)
19 \( 1 - 1.94e4iT - 1.69e10T^{2} \)
23 \( 1 + 2.37e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.28e5T + 5.00e11T^{2} \)
31 \( 1 - 6.79e4iT - 8.52e11T^{2} \)
37 \( 1 + 3.47e6T + 3.51e12T^{2} \)
41 \( 1 + 2.14e6T + 7.98e12T^{2} \)
43 \( 1 - 5.92e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.62e6iT - 2.38e13T^{2} \)
53 \( 1 + 8.24e5T + 6.22e13T^{2} \)
59 \( 1 + 3.72e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.47e7T + 1.91e14T^{2} \)
67 \( 1 + 1.52e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.19e6iT - 6.45e14T^{2} \)
73 \( 1 + 5.72e6T + 8.06e14T^{2} \)
79 \( 1 + 3.59e7iT - 1.51e15T^{2} \)
83 \( 1 + 5.19e7iT - 2.25e15T^{2} \)
89 \( 1 - 8.33e7T + 3.93e15T^{2} \)
97 \( 1 - 1.20e8T + 7.83e15T^{2} \)
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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

−14.36383764308742307887130874321, −13.37210605020845852451547283612, −12.06445528929317942738641893720, −10.91900450625161060874570279613, −8.732066381144014821980541357217, −7.67581711653356857228475449892, −6.34903529750758046397638789546, −4.58065403281561724866601500377, −3.32110378641194018008605617639, −0.11020138669429588310646281962, 2.06667444578291569518660232734, 3.81334955851094071078320210871, 5.13972516166896499660690395725, 7.04633175919398158631470447999, 8.889668982782506414303774776988, 10.29334724703011766469428241593, 11.70223215753488133288133480616, 12.31754605253551804231961851868, 13.65017761104597776898732760040, 15.20938211792639549671811020406

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