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Properties

Label 2-6e2-36.23-c3-0-7
Degree $2$
Conductor $36$
Sign $0.811 + 0.584i$
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  + (−2.52 − 1.27i)2-s + (5.18 − 0.355i)3-s + (4.74 + 6.44i)4-s + (−1.23 − 0.715i)5-s + (−13.5 − 5.71i)6-s + (23.8 − 13.7i)7-s + (−3.76 − 22.3i)8-s + (26.7 − 3.68i)9-s + (2.21 + 3.38i)10-s + (−11.1 − 19.2i)11-s + (26.8 + 31.7i)12-s + (−34.5 + 59.9i)13-s + (−77.8 + 4.33i)14-s + (−6.67 − 3.26i)15-s + (−18.9 + 61.1i)16-s + 31.4i·17-s + ⋯
L(s)  = 1  + (−0.892 − 0.451i)2-s + (0.997 − 0.0683i)3-s + (0.593 + 0.805i)4-s + (−0.110 − 0.0639i)5-s + (−0.921 − 0.388i)6-s + (1.28 − 0.744i)7-s + (−0.166 − 0.986i)8-s + (0.990 − 0.136i)9-s + (0.0700 + 0.107i)10-s + (−0.304 − 0.527i)11-s + (0.646 + 0.762i)12-s + (−0.738 + 1.27i)13-s + (−1.48 + 0.0828i)14-s + (−0.114 − 0.0562i)15-s + (−0.296 + 0.955i)16-s + 0.448i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(2.12406\)
Root analytic conductor: \(1.45741\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :3/2),\ 0.811 + 0.584i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.14732 - 0.370345i\)
\(L(\frac12)\) \(\approx\) \(1.14732 - 0.370345i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.52 + 1.27i)T \)
3 \( 1 + (-5.18 + 0.355i)T \)
good5 \( 1 + (1.23 + 0.715i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-23.8 + 13.7i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (11.1 + 19.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (34.5 - 59.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 31.4iT - 4.91e3T^{2} \)
19 \( 1 + 11.4iT - 6.85e3T^{2} \)
23 \( 1 + (72.6 - 125. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (93.6 - 54.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (102. + 59.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 300.T + 5.06e4T^{2} \)
41 \( 1 + (-344. - 199. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-173. + 100. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (151. + 262. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 243. iT - 1.48e5T^{2} \)
59 \( 1 + (41.9 - 72.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-199. - 345. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (307. + 177. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 866.T + 3.57e5T^{2} \)
73 \( 1 - 64.6T + 3.89e5T^{2} \)
79 \( 1 + (-354. + 204. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-79.8 - 138. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 + (-700. - 1.21e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−16.01297496034819337684455059957, −14.59061641362701669889543403904, −13.56905444583349277904484873923, −11.97015284490549748817398626630, −10.75963574502542049150516088910, −9.413292130602089575019265292180, −8.174828041542768363089803534115, −7.29185466463485211408010671890, −4.04893392898809410912145229624, −1.85488749605626474199913172940, 2.22479002996772461703893723506, 5.19725742416221891397775755261, 7.49482998315104966781236146383, 8.253052561785771743379805323866, 9.523858942469713107600218087155, 10.79895714838490830949908382857, 12.41515087261441104346291662311, 14.31144009482200591153583751513, 15.00029734870511690501046489555, 15.82760282236134852113391834253

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