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Properties

Label 2-6e2-36.23-c3-0-2
Degree $2$
Conductor $36$
Sign $0.802 - 0.596i$
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  + (−1.38 − 2.46i)2-s + (−1.42 + 4.99i)3-s + (−4.17 + 6.82i)4-s + (14.6 + 8.45i)5-s + (14.3 − 3.38i)6-s + (−3.08 + 1.78i)7-s + (22.6 + 0.866i)8-s + (−22.9 − 14.2i)9-s + (0.610 − 47.8i)10-s + (25.0 + 43.4i)11-s + (−28.1 − 30.6i)12-s + (−18.9 + 32.9i)13-s + (8.67 + 5.15i)14-s + (−63.2 + 61.0i)15-s + (−29.1 − 56.9i)16-s − 84.3i·17-s + ⋯
L(s)  = 1  + (−0.488 − 0.872i)2-s + (−0.275 + 0.961i)3-s + (−0.521 + 0.852i)4-s + (1.31 + 0.756i)5-s + (0.973 − 0.229i)6-s + (−0.166 + 0.0963i)7-s + (0.999 + 0.0382i)8-s + (−0.848 − 0.529i)9-s + (0.0193 − 1.51i)10-s + (0.687 + 1.19i)11-s + (−0.676 − 0.736i)12-s + (−0.405 + 0.701i)13-s + (0.165 + 0.0984i)14-s + (−1.08 + 1.05i)15-s + (−0.455 − 0.890i)16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.802 - 0.596i$
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: \((2,\ 36,\ (\ :3/2),\ 0.802 - 0.596i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.959584 + 0.317280i\)
\(L(\frac12)\) \(\approx\) \(0.959584 + 0.317280i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 2.46i)T \)
3 \( 1 + (1.42 - 4.99i)T \)
good5 \( 1 + (-14.6 - 8.45i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (3.08 - 1.78i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-25.0 - 43.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (18.9 - 32.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 84.3iT - 4.91e3T^{2} \)
19 \( 1 + 62.9iT - 6.85e3T^{2} \)
23 \( 1 + (-37.6 + 65.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-105. + 60.9i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (17.2 + 9.97i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 17.7T + 5.06e4T^{2} \)
41 \( 1 + (-299. - 172. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-113. + 65.3i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (153. + 265. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 479. iT - 1.48e5T^{2} \)
59 \( 1 + (245. - 425. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (49.9 + 86.4i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (536. + 309. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 254.T + 3.57e5T^{2} \)
73 \( 1 - 100.T + 3.89e5T^{2} \)
79 \( 1 + (-856. + 494. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-251. - 436. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.01e3iT - 7.04e5T^{2} \)
97 \( 1 + (503. + 872. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−16.44708902808231566415560749306, −14.76042111446063902575644085288, −13.74805937437256989677561764838, −12.10879268173700124986387256977, −10.92010643376261291556324010948, −9.713640726664929027620174987666, −9.331954070308053389241586882915, −6.77239480369149172293538434279, −4.68027462035480779050176741586, −2.57187555332940204792177224237, 1.25893256018659242751237025999, 5.55232657340776149257924320922, 6.30512096096777026969126279804, 8.064653382054780076167667678831, 9.190202122936365083039954147638, 10.64870808371955661913658716578, 12.63222001219344688775054287322, 13.58194978168124204373274571900, 14.46266794815274160496186650347, 16.34674686393460628346763749966

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