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Properties

Label 2-6e2-36.11-c3-0-1
Degree $2$
Conductor $36$
Sign $-0.969 - 0.243i$
Analytic cond. $2.12406$
Root an. cond. $1.45741$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.664 + 2.74i)2-s + (−5.00 − 1.39i)3-s + (−7.11 + 3.65i)4-s + (−14.2 + 8.25i)5-s + (0.524 − 14.6i)6-s + (19.2 + 11.1i)7-s + (−14.7 − 17.1i)8-s + (23.0 + 14.0i)9-s + (−32.1 − 33.8i)10-s + (−6.37 + 11.0i)11-s + (40.7 − 8.31i)12-s + (−11.1 − 19.3i)13-s + (−17.7 + 60.3i)14-s + (83.0 − 21.2i)15-s + (37.3 − 51.9i)16-s + 117. i·17-s + ⋯
L(s)  = 1  + (0.234 + 0.972i)2-s + (−0.963 − 0.269i)3-s + (−0.889 + 0.456i)4-s + (−1.27 + 0.737i)5-s + (0.0356 − 0.999i)6-s + (1.04 + 0.600i)7-s + (−0.652 − 0.757i)8-s + (0.854 + 0.518i)9-s + (−1.01 − 1.06i)10-s + (−0.174 + 0.302i)11-s + (0.979 − 0.199i)12-s + (−0.238 − 0.413i)13-s + (−0.339 + 1.15i)14-s + (1.42 − 0.366i)15-s + (0.583 − 0.812i)16-s + 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.969 - 0.243i$
Analytic conductor: \(2.12406\)
Root analytic conductor: \(1.45741\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :3/2),\ -0.969 - 0.243i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0818011 + 0.661946i\)
\(L(\frac12)\) \(\approx\) \(0.0818011 + 0.661946i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.664 - 2.74i)T \)
3 \( 1 + (5.00 + 1.39i)T \)
good5 \( 1 + (14.2 - 8.25i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-19.2 - 11.1i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (6.37 - 11.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (11.1 + 19.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 117. iT - 4.91e3T^{2} \)
19 \( 1 - 27.7iT - 6.85e3T^{2} \)
23 \( 1 + (17.5 + 30.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-1.01 - 0.584i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (119. - 68.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 233.T + 5.06e4T^{2} \)
41 \( 1 + (-13.2 + 7.65i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-361. - 208. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-116. + 201. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 180. iT - 1.48e5T^{2} \)
59 \( 1 + (-313. - 543. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (382. - 661. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (113. - 65.5i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 22.6T + 3.57e5T^{2} \)
73 \( 1 - 387.T + 3.89e5T^{2} \)
79 \( 1 + (486. + 280. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-342. + 592. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 278. iT - 7.04e5T^{2} \)
97 \( 1 + (264. - 458. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−16.40422537644252883120081386205, −15.23333654660057830552612395099, −14.70311649037363177977308176558, −12.76867322389275169278518335716, −11.83011693329616647611742318148, −10.60736171011384484036327302777, −8.229714554329184921620515713768, −7.35145047842021930649147567205, −5.82413916217688635861518542565, −4.28228792560961888065653789515, 0.64127928453103219404222147034, 4.17981495027682821456903438579, 5.06830426414808823415356381146, 7.65990650357583247969466313885, 9.338869428436131588829474763024, 11.05972119897948146264657882147, 11.53076180054749649721002073490, 12.55141004378210944423262140971, 14.04174379099480119610924565998, 15.53500124178934736760496101293

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