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Properties

Label 2-6e2-9.4-c5-0-2
Degree $2$
Conductor $36$
Sign $0.779 - 0.625i$
Analytic cond. $5.77381$
Root an. cond. $2.40287$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.64 + 13.5i)3-s + (40.7 − 70.5i)5-s + (89.6 + 155. i)7-s + (−126. + 207. i)9-s + (250. + 433. i)11-s + (275. − 476. i)13-s + (1.26e3 + 13.9i)15-s + 753.·17-s − 2.57e3·19-s + (−1.42e3 + 2.40e3i)21-s + (1.37e3 − 2.37e3i)23-s + (−1.75e3 − 3.03e3i)25-s + (−3.78e3 − 124. i)27-s + (−1.95e3 − 3.38e3i)29-s + (1.55e3 − 2.68e3i)31-s + ⋯
L(s)  = 1  + (0.490 + 0.871i)3-s + (0.728 − 1.26i)5-s + (0.691 + 1.19i)7-s + (−0.518 + 0.854i)9-s + (0.623 + 1.08i)11-s + (0.451 − 0.782i)13-s + (1.45 + 0.0159i)15-s + 0.632·17-s − 1.63·19-s + (−0.704 + 1.18i)21-s + (0.541 − 0.937i)23-s + (−0.561 − 0.972i)25-s + (−0.999 − 0.0329i)27-s + (−0.431 − 0.747i)29-s + (0.290 − 0.502i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.779 - 0.625i$
Analytic conductor: \(5.77381\)
Root analytic conductor: \(2.40287\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :5/2),\ 0.779 - 0.625i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.98362 + 0.697428i\)
\(L(\frac12)\) \(\approx\) \(1.98362 + 0.697428i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-7.64 - 13.5i)T \)
good5 \( 1 + (-40.7 + 70.5i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-89.6 - 155. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-250. - 433. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (-275. + 476. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 - 753.T + 1.41e6T^{2} \)
19 \( 1 + 2.57e3T + 2.47e6T^{2} \)
23 \( 1 + (-1.37e3 + 2.37e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.95e3 + 3.38e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.55e3 + 2.68e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 9.56e3T + 6.93e7T^{2} \)
41 \( 1 + (1.11e3 - 1.92e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (7.14e3 + 1.23e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-3.23e3 - 5.60e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 1.36e4T + 4.18e8T^{2} \)
59 \( 1 + (2.85e3 - 4.94e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-5.89e3 - 1.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.77e3 + 3.06e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 5.84e4T + 1.80e9T^{2} \)
73 \( 1 + 6.01e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.78e4 - 4.81e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (1.99e4 + 3.46e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 1.03e5T + 5.58e9T^{2} \)
97 \( 1 + (-8.29e4 - 1.43e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−15.36015744176912739988026274439, −14.69168142511005804019403666102, −13.13444375485960110767123556759, −12.06003396113525225915620431891, −10.31174119690893952902747036482, −9.054144377661144835443863537221, −8.394797935889791402072747964146, −5.64516881232223088269416423564, −4.53111923151662121292449313957, −2.04472393262952478261520453737, 1.54543079295207847757507091375, 3.48232017417218388007299499696, 6.28805172064575252938166177173, 7.22141386575042349625421124387, 8.744888094174782965228441461047, 10.50567069520440760847515319226, 11.46748235530457100686060397842, 13.35850897048868858020629634826, 14.11537610866230569278724504199, 14.69599552832621335626086174157

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