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Properties

Label 2-6e2-4.3-c8-0-8
Degree $2$
Conductor $36$
Sign $0.937 + 0.347i$
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  + (2.82 − 15.7i)2-s + (−240. − 88.8i)4-s + 1.03e3·5-s + 3.62e3i·7-s + (−2.07e3 + 3.53e3i)8-s + (2.91e3 − 1.62e4i)10-s + 1.58e4i·11-s + 3.07e4·13-s + (5.70e4 + 1.02e4i)14-s + (4.97e4 + 4.26e4i)16-s + 1.19e4·17-s − 1.70e5i·19-s + (−2.47e5 − 9.17e4i)20-s + (2.50e5 + 4.48e4i)22-s + 3.56e4i·23-s + ⋯
L(s)  = 1  + (0.176 − 0.984i)2-s + (−0.937 − 0.347i)4-s + 1.65·5-s + 1.50i·7-s + (−0.506 + 0.861i)8-s + (0.291 − 1.62i)10-s + 1.08i·11-s + 1.07·13-s + (1.48 + 0.265i)14-s + (0.759 + 0.651i)16-s + 0.143·17-s − 1.30i·19-s + (−1.54 − 0.573i)20-s + (1.06 + 0.191i)22-s + 0.127i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.937 + 0.347i$
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: \((2,\ 36,\ (\ :4),\ 0.937 + 0.347i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.34147 - 0.419379i\)
\(L(\frac12)\) \(\approx\) \(2.34147 - 0.419379i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 + 15.7i)T \)
3 \( 1 \)
good5 \( 1 - 1.03e3T + 3.90e5T^{2} \)
7 \( 1 - 3.62e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.58e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.07e4T + 8.15e8T^{2} \)
17 \( 1 - 1.19e4T + 6.97e9T^{2} \)
19 \( 1 + 1.70e5iT - 1.69e10T^{2} \)
23 \( 1 - 3.56e4iT - 7.83e10T^{2} \)
29 \( 1 + 1.04e5T + 5.00e11T^{2} \)
31 \( 1 - 5.89e5iT - 8.52e11T^{2} \)
37 \( 1 + 1.13e6T + 3.51e12T^{2} \)
41 \( 1 - 3.50e6T + 7.98e12T^{2} \)
43 \( 1 + 2.85e6iT - 1.16e13T^{2} \)
47 \( 1 - 7.54e6iT - 2.38e13T^{2} \)
53 \( 1 + 9.14e6T + 6.22e13T^{2} \)
59 \( 1 - 8.04e6iT - 1.46e14T^{2} \)
61 \( 1 - 6.68e6T + 1.91e14T^{2} \)
67 \( 1 + 3.09e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.54e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.84e7T + 8.06e14T^{2} \)
79 \( 1 + 1.78e7iT - 1.51e15T^{2} \)
83 \( 1 + 8.08e7iT - 2.25e15T^{2} \)
89 \( 1 - 4.79e7T + 3.93e15T^{2} \)
97 \( 1 + 4.98e7T + 7.83e15T^{2} \)
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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

−14.29669365192505320084637992623, −13.20590881361587094647408429573, −12.33007461157221379303183286557, −10.88061581515678911638392538686, −9.544502469132237091791388103236, −8.910741106882295926363040336361, −6.12035962747334700602978979511, −5.04466279478916239934779748033, −2.66426864900564345266167890927, −1.64364084852318195892149055639, 1.05291050578426973150522224119, 3.74773180525677618640819054155, 5.62538808825914134009412563413, 6.54838244007709245339575312949, 8.168506366281900349736648059820, 9.614670998752702256288403407992, 10.69547862208188115710757372819, 13.06003157423353562111711817215, 13.76971541605987892305443300516, 14.35359940632643530178328998244

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