Properties

Label 2-6e2-36.11-c5-0-27
Degree $2$
Conductor $36$
Sign $-0.973 - 0.230i$
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  + (3.65 − 4.31i)2-s + (−9.73 − 12.1i)3-s + (−5.21 − 31.5i)4-s + (−51.8 + 29.9i)5-s + (−88.1 − 2.54i)6-s + (33.8 + 19.5i)7-s + (−155. − 93.0i)8-s + (−53.3 + 237. i)9-s + (−60.6 + 333. i)10-s + (108. − 187. i)11-s + (−333. + 370. i)12-s + (−532. − 921. i)13-s + (208. − 74.5i)14-s + (869. + 339. i)15-s + (−969. + 329. i)16-s + 315. i·17-s + ⋯
L(s)  = 1  + (0.646 − 0.762i)2-s + (−0.624 − 0.780i)3-s + (−0.162 − 0.986i)4-s + (−0.927 + 0.535i)5-s + (−0.999 − 0.0288i)6-s + (0.261 + 0.150i)7-s + (−0.857 − 0.514i)8-s + (−0.219 + 0.975i)9-s + (−0.191 + 1.05i)10-s + (0.270 − 0.468i)11-s + (−0.668 + 0.743i)12-s + (−0.873 − 1.51i)13-s + (0.284 − 0.101i)14-s + (0.997 + 0.389i)15-s + (−0.946 + 0.321i)16-s + 0.264i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $-0.973 - 0.230i$
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: \((2,\ 36,\ (\ :5/2),\ -0.973 - 0.230i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.113070 + 0.969502i\)
\(L(\frac12)\) \(\approx\) \(0.113070 + 0.969502i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.65 + 4.31i)T \)
3 \( 1 + (9.73 + 12.1i)T \)
good5 \( 1 + (51.8 - 29.9i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-33.8 - 19.5i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-108. + 187. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (532. + 921. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 315. iT - 1.41e6T^{2} \)
19 \( 1 + 1.88e3iT - 2.47e6T^{2} \)
23 \( 1 + (1.20e3 + 2.07e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-5.18e3 - 2.99e3i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-6.94e3 + 4.01e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 5.24e3T + 6.93e7T^{2} \)
41 \( 1 + (7.58e3 - 4.38e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (8.78e3 + 5.06e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (7.27e3 - 1.25e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 1.96e4iT - 4.18e8T^{2} \)
59 \( 1 + (909. + 1.57e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.44e4 + 4.22e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (3.70e4 - 2.14e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 2.94e4T + 1.80e9T^{2} \)
73 \( 1 - 3.80e4T + 2.07e9T^{2} \)
79 \( 1 + (-1.28e4 - 7.42e3i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-6.01e4 + 1.04e5i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 3.54e4iT - 5.58e9T^{2} \)
97 \( 1 + (-1.49e3 + 2.58e3i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−14.65666455753080271162584220026, −13.30218165833927176438233281957, −12.21593678307848349139797100990, −11.38252469911641904430155204430, −10.37060545423390316526276619011, −8.074518426020705697931976618740, −6.50567884680329799564932078704, −4.91169240490989232058587897706, −2.83268871261777833339164328860, −0.49926396739770510491628094869, 4.01765454126841187309193162291, 4.87032820512476270651051916974, 6.65258374471319645255728382698, 8.183568680615044522090803597151, 9.704124780040011589725270562603, 11.86010547013293090555014053355, 12.01599640288458103370304903458, 14.02967935859505539090887464589, 15.07348946507478875506252961568, 16.11156475456765446305805817286

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