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Properties

Label 2-6e2-36.11-c3-0-0
Degree $2$
Conductor $36$
Sign $-0.935 + 0.352i$
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  + (−1.93 + 2.06i)2-s + (−2.72 + 4.42i)3-s + (−0.538 − 7.98i)4-s + (−4.71 + 2.72i)5-s + (−3.88 − 14.1i)6-s + (−20.9 − 12.0i)7-s + (17.5 + 14.3i)8-s + (−12.1 − 24.1i)9-s + (3.48 − 14.9i)10-s + (−25.3 + 43.9i)11-s + (36.7 + 19.3i)12-s + (25.0 + 43.4i)13-s + (65.4 − 19.9i)14-s + (0.794 − 28.2i)15-s + (−63.4 + 8.60i)16-s + 51.7i·17-s + ⋯
L(s)  = 1  + (−0.682 + 0.730i)2-s + (−0.524 + 0.851i)3-s + (−0.0673 − 0.997i)4-s + (−0.421 + 0.243i)5-s + (−0.264 − 0.964i)6-s + (−1.13 − 0.652i)7-s + (0.774 + 0.632i)8-s + (−0.450 − 0.892i)9-s + (0.110 − 0.474i)10-s + (−0.696 + 1.20i)11-s + (0.885 + 0.465i)12-s + (0.535 + 0.927i)13-s + (1.24 − 0.380i)14-s + (0.0136 − 0.486i)15-s + (−0.990 + 0.134i)16-s + 0.737i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.935 + 0.352i$
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: \((2,\ 36,\ (\ :3/2),\ -0.935 + 0.352i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0555309 - 0.304882i\)
\(L(\frac12)\) \(\approx\) \(0.0555309 - 0.304882i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.93 - 2.06i)T \)
3 \( 1 + (2.72 - 4.42i)T \)
good5 \( 1 + (4.71 - 2.72i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (20.9 + 12.0i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (25.3 - 43.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-25.0 - 43.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 51.7iT - 4.91e3T^{2} \)
19 \( 1 + 27.9iT - 6.85e3T^{2} \)
23 \( 1 + (-3.93 - 6.81i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (212. + 122. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-51.4 + 29.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 + (146. - 84.7i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-284. - 164. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-47.9 + 83.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 300. iT - 1.48e5T^{2} \)
59 \( 1 + (113. + 196. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-173. + 300. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (904. - 522. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 243.T + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + (-530. - 306. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (283. - 490. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 212. iT - 7.04e5T^{2} \)
97 \( 1 + (-234. + 405. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−16.55455491104916380213162540081, −15.65647405017199240206735181950, −14.86990302310488347081431792968, −13.18321890663053591183610084667, −11.31024456863568457517352836645, −10.16391345017061631686587758990, −9.319429666939917262973024234055, −7.40093574253446951913834707528, −6.13136566912813737862881406485, −4.23765574870495837312528002842, 0.34566135393146410948046424907, 2.99474231916670549503160985903, 5.88589148045333968865502167088, 7.66447510178700735150915512008, 8.827662665977587359593725345133, 10.51349271847645844839984191923, 11.66078852662509084760873967127, 12.71635961642148409160540172059, 13.43920492372881314959144106574, 15.93911167780767478165187248514

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