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Properties

Label 2-6e2-9.7-c5-0-3
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: $\chi_{36} (25, \cdot )$
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

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

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