Properties

Degree $2$
Conductor $36$
Sign $0.0411 - 0.999i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.67 + 12.9i)3-s + (13.1 + 22.7i)5-s + (−31.6 + 54.7i)7-s + (−92.4 + 224. i)9-s + (−49.1 + 85.0i)11-s + (369. + 639. i)13-s + (−181. + 368. i)15-s + 250.·17-s + 1.10e3·19-s + (−983. + 65.6i)21-s + (−2.20e3 − 3.81e3i)23-s + (1.21e3 − 2.10e3i)25-s + (−3.71e3 + 752. i)27-s + (3.94e3 − 6.82e3i)29-s + (2.30e3 + 3.99e3i)31-s + ⋯
L(s)  = 1  + (0.556 + 0.830i)3-s + (0.235 + 0.407i)5-s + (−0.243 + 0.422i)7-s + (−0.380 + 0.924i)9-s + (−0.122 + 0.211i)11-s + (0.605 + 1.04i)13-s + (−0.207 + 0.422i)15-s + 0.209·17-s + 0.700·19-s + (−0.486 + 0.0325i)21-s + (−0.869 − 1.50i)23-s + (0.389 − 0.674i)25-s + (−0.980 + 0.198i)27-s + (0.870 − 1.50i)29-s + (0.430 + 0.746i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0411 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0411 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.0411 - 0.999i$
Motivic weight: \(5\)
Character: $\chi_{36} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :5/2),\ 0.0411 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.30081 + 1.24832i\)
\(L(\frac12)\) \(\approx\) \(1.30081 + 1.24832i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-8.67 - 12.9i)T \)
good5 \( 1 + (-13.1 - 22.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (31.6 - 54.7i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (49.1 - 85.0i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-369. - 639. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 250.T + 1.41e6T^{2} \)
19 \( 1 - 1.10e3T + 2.47e6T^{2} \)
23 \( 1 + (2.20e3 + 3.81e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-3.94e3 + 6.82e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-2.30e3 - 3.99e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 1.18e4T + 6.93e7T^{2} \)
41 \( 1 + (5.04e3 + 8.73e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (3.51e3 - 6.09e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (7.45e3 - 1.29e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 2.24e4T + 4.18e8T^{2} \)
59 \( 1 + (5.40e3 + 9.36e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-594. + 1.02e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.95e4 + 5.12e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.43e4T + 1.80e9T^{2} \)
73 \( 1 + 5.30e4T + 2.07e9T^{2} \)
79 \( 1 + (-1.86e4 + 3.23e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (6.04e4 - 1.04e5i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 9.78e4T + 5.58e9T^{2} \)
97 \( 1 + (5.33e4 - 9.24e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−15.75025581234880966458482642581, −14.51395758070953082120335557795, −13.70516372744239013371842026244, −11.98099203025854396137867103518, −10.54292191411282057544008506143, −9.490478520550486538833384230921, −8.231222412111198096909404726809, −6.28876579323792326674770385210, −4.39235318511634759413500986955, −2.60499975076277627794828287755, 1.11381030846305132922865692063, 3.29596491764428199896444655960, 5.72707840240066854354321524310, 7.36056279768466671753677016644, 8.552189316899365146969877862903, 9.966580975295799278814681356532, 11.68475712178476635174147812349, 13.05721574468693409594422001609, 13.63732317415724076902250734545, 15.02555148321566480056659180644

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