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Properties

Label 2-6e2-4.3-c6-0-6
Degree $2$
Conductor $36$
Sign $0.108 - 0.994i$
Analytic cond. $8.28194$
Root an. cond. $2.87783$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.33 + 5.95i)2-s + (−6.97 + 63.6i)4-s + 212.·5-s − 87.6i·7-s + (−416. + 298. i)8-s + (1.13e3 + 1.26e3i)10-s + 2.01e3i·11-s − 27.7·13-s + (521. − 467. i)14-s + (−3.99e3 − 887. i)16-s + 1.07e3·17-s − 2.47e3i·19-s + (−1.48e3 + 1.35e4i)20-s + (−1.20e4 + 1.07e4i)22-s − 1.39e4i·23-s + ⋯
L(s)  = 1  + (0.667 + 0.744i)2-s + (−0.108 + 0.994i)4-s + 1.69·5-s − 0.255i·7-s + (−0.812 + 0.582i)8-s + (1.13 + 1.26i)10-s + 1.51i·11-s − 0.0126·13-s + (0.190 − 0.170i)14-s + (−0.976 − 0.216i)16-s + 0.218·17-s − 0.360i·19-s + (−0.185 + 1.68i)20-s + (−1.12 + 1.01i)22-s − 1.14i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.108 - 0.994i$
Analytic conductor: \(8.28194\)
Root analytic conductor: \(2.87783\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :3),\ 0.108 - 0.994i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.11737 + 1.89797i\)
\(L(\frac12)\) \(\approx\) \(2.11737 + 1.89797i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.33 - 5.95i)T \)
3 \( 1 \)
good5 \( 1 - 212.T + 1.56e4T^{2} \)
7 \( 1 + 87.6iT - 1.17e5T^{2} \)
11 \( 1 - 2.01e3iT - 1.77e6T^{2} \)
13 \( 1 + 27.7T + 4.82e6T^{2} \)
17 \( 1 - 1.07e3T + 2.41e7T^{2} \)
19 \( 1 + 2.47e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.39e4iT - 1.48e8T^{2} \)
29 \( 1 - 6.45e3T + 5.94e8T^{2} \)
31 \( 1 + 3.87e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.96e4T + 2.56e9T^{2} \)
41 \( 1 - 5.08e4T + 4.75e9T^{2} \)
43 \( 1 + 8.35e4iT - 6.32e9T^{2} \)
47 \( 1 + 2.87e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.49e5T + 2.21e10T^{2} \)
59 \( 1 - 3.66e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.00e5T + 5.15e10T^{2} \)
67 \( 1 + 1.00e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.93e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.39e5T + 1.51e11T^{2} \)
79 \( 1 + 1.99e5iT - 2.43e11T^{2} \)
83 \( 1 - 4.70e4iT - 3.26e11T^{2} \)
89 \( 1 + 5.52e5T + 4.96e11T^{2} \)
97 \( 1 - 3.32e5T + 8.32e11T^{2} \)
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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

−15.21494815271780067615389120256, −14.22781530917013997849828228196, −13.29781427280612881068544889594, −12.32300399970817583285069506852, −10.28644658579122603080712495220, −9.091248034124188620081214039851, −7.22135114757217624666468083479, −6.01220749855602524318758409353, −4.65750454709897162119693111643, −2.29256597851472634848161538925, 1.43560951197484635220951772138, 3.05990238279497896502761823117, 5.36090453123850579388315331886, 6.21325948065240406300773218147, 8.951890023030222947574584762158, 10.05266562861153411370285292811, 11.17571134635218237646427192406, 12.68263356396470726747605058611, 13.75884687575519528698927827325, 14.28373055132920105786622753173

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