Properties

Label 2-6e2-36.11-c5-0-7
Degree $2$
Conductor $36$
Sign $-0.999 + 0.0415i$
Analytic cond. $5.77381$
Root an. cond. $2.40287$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.66 + 4.99i)2-s + (−5.80 + 14.4i)3-s + (−17.8 + 26.5i)4-s + (3.68 − 2.12i)5-s + (−87.6 + 9.50i)6-s + (13.9 + 8.04i)7-s + (−180. − 18.3i)8-s + (−175. − 168. i)9-s + (20.4 + 12.7i)10-s + (−43.3 + 75.0i)11-s + (−280. − 412. i)12-s + (162. + 281. i)13-s + (−3.07 + 90.9i)14-s + (9.36 + 65.5i)15-s + (−387. − 947. i)16-s + 1.86e3i·17-s + ⋯
L(s)  = 1  + (0.470 + 0.882i)2-s + (−0.372 + 0.928i)3-s + (−0.557 + 0.830i)4-s + (0.0658 − 0.0380i)5-s + (−0.994 + 0.107i)6-s + (0.107 + 0.0620i)7-s + (−0.994 − 0.101i)8-s + (−0.722 − 0.691i)9-s + (0.0645 + 0.0402i)10-s + (−0.107 + 0.186i)11-s + (−0.562 − 0.826i)12-s + (0.266 + 0.461i)13-s + (−0.00419 + 0.124i)14-s + (0.0107 + 0.0752i)15-s + (−0.378 − 0.925i)16-s + 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0415i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.999 + 0.0415i$
Analytic conductor: \(5.77381\)
Root analytic conductor: \(2.40287\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :5/2),\ -0.999 + 0.0415i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0284208 - 1.36722i\)
\(L(\frac12)\) \(\approx\) \(0.0284208 - 1.36722i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.66 - 4.99i)T \)
3 \( 1 + (5.80 - 14.4i)T \)
good5 \( 1 + (-3.68 + 2.12i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-13.9 - 8.04i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (43.3 - 75.0i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-162. - 281. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.86e3iT - 1.41e6T^{2} \)
19 \( 1 + 416. iT - 2.47e6T^{2} \)
23 \( 1 + (-1.29e3 - 2.24e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-4.34e3 - 2.50e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-6.39e3 + 3.69e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 1.05e4T + 6.93e7T^{2} \)
41 \( 1 + (-1.32e4 + 7.62e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (6.79e3 + 3.92e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.42e4 - 2.46e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 5.48e3iT - 4.18e8T^{2} \)
59 \( 1 + (-1.90e3 - 3.29e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.65e4 + 2.87e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-5.75e4 + 3.32e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 4.35e4T + 1.80e9T^{2} \)
73 \( 1 - 2.29e4T + 2.07e9T^{2} \)
79 \( 1 + (-3.19e4 - 1.84e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (2.75e4 - 4.76e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 4.09e4iT - 5.58e9T^{2} \)
97 \( 1 + (-2.33e4 + 4.04e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−15.81941591567973276260029364856, −15.13031476066567710459477758462, −13.95976764741833749750095596305, −12.55812530387646323566533061392, −11.19608520818036391432732028798, −9.601865712497367996488993270101, −8.321198392863914233131185377000, −6.46731236539712053691962284808, −5.13799253017541438558595292113, −3.72182588163292410545551104608, 0.74653175756927280905625769307, 2.68261141737635042661838152933, 5.00885817314344463013199362363, 6.51188015558454640120995990733, 8.379424251236861650564147003110, 10.16562994543855874683212481088, 11.42649359190921897862017997780, 12.32774776893539893340228287904, 13.49530761789160013722098115485, 14.27607807186286962364338100356

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