Properties

Degree $2$
Conductor $36$
Sign $-0.131 + 0.991i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.35 − 1.83i)2-s + (6.54 − 14.1i)3-s + (25.2 + 19.6i)4-s + (36.0 − 20.8i)5-s + (−61.0 + 63.6i)6-s + (117. + 67.7i)7-s + (−99.1 − 151. i)8-s + (−157. − 185. i)9-s + (−231. + 45.2i)10-s + (144. − 249. i)11-s + (443. − 228. i)12-s + (−399. − 692. i)13-s + (−503. − 577. i)14-s + (−58.3 − 646. i)15-s + (252. + 992. i)16-s − 1.42e3i·17-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)2-s + (0.420 − 0.907i)3-s + (0.789 + 0.613i)4-s + (0.645 − 0.372i)5-s + (−0.691 + 0.722i)6-s + (0.904 + 0.522i)7-s + (−0.547 − 0.836i)8-s + (−0.647 − 0.762i)9-s + (−0.730 + 0.143i)10-s + (0.359 − 0.622i)11-s + (0.888 − 0.458i)12-s + (−0.655 − 1.13i)13-s + (−0.686 − 0.787i)14-s + (−0.0670 − 0.741i)15-s + (0.246 + 0.969i)16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.131 + 0.991i$
Motivic weight: \(5\)
Character: $\chi_{36} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :5/2),\ -0.131 + 0.991i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.876375 - 1.00001i\)
\(L(\frac12)\) \(\approx\) \(0.876375 - 1.00001i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.35 + 1.83i)T \)
3 \( 1 + (-6.54 + 14.1i)T \)
good5 \( 1 + (-36.0 + 20.8i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-117. - 67.7i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-144. + 249. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (399. + 692. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + 1.42e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.19e3iT - 2.47e6T^{2} \)
23 \( 1 + (276. + 478. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-2.03e3 - 1.17e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-8.14e3 + 4.70e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 2.86e3T + 6.93e7T^{2} \)
41 \( 1 + (2.49e3 - 1.44e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-5.63e3 - 3.25e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (3.57e3 - 6.19e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 1.98e4iT - 4.18e8T^{2} \)
59 \( 1 + (-2.46e4 - 4.26e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.93e4 - 3.34e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.70e4 + 2.13e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.94e4T + 1.80e9T^{2} \)
73 \( 1 - 5.26e4T + 2.07e9T^{2} \)
79 \( 1 + (1.30e3 + 751. i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (4.53e4 - 7.85e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 1.72e4iT - 5.58e9T^{2} \)
97 \( 1 + (-4.25e4 + 7.37e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−15.09829834664596536447551659913, −13.80986921030569195454349952143, −12.44717126772976001816112246190, −11.55281766485377126801739267222, −9.834621116057036177112622813563, −8.575390302008903732943484810724, −7.66286116867839785614727470072, −5.87129269748786453913503720941, −2.63786191342002923349916527166, −1.09853216433706749241420089277, 2.08833960394597986074292635524, 4.71006511770505662724768246934, 6.70270268380786117236279632036, 8.258753774415403548411818020482, 9.524783553944561346271865712695, 10.42203328558762721161681580756, 11.54481021435696806136620301606, 14.01513383773434503419712379083, 14.68469417703383036465394947432, 15.78405956101477810928921538964

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