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Properties

Label 2-37-37.27-c5-0-2
Degree $2$
Conductor $37$
Sign $0.944 - 0.327i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
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  + (−7.60 − 4.39i)2-s + (−8.72 − 15.1i)3-s + (22.5 + 39.0i)4-s + (−72.9 + 42.1i)5-s + 153. i·6-s + (−77.9 − 134. i)7-s − 114. i·8-s + (−30.7 + 53.3i)9-s + 739.·10-s + 403.·11-s + (393. − 681. i)12-s + (−257. + 148. i)13-s + 1.36e3i·14-s + (1.27e3 + 735. i)15-s + (216. − 375. i)16-s + (−468. − 270. i)17-s + ⋯
L(s)  = 1  + (−1.34 − 0.776i)2-s + (−0.559 − 0.969i)3-s + (0.704 + 1.22i)4-s + (−1.30 + 0.753i)5-s + 1.73i·6-s + (−0.601 − 1.04i)7-s − 0.634i·8-s + (−0.126 + 0.219i)9-s + 2.33·10-s + 1.00·11-s + (0.788 − 1.36i)12-s + (−0.422 + 0.244i)13-s + 1.86i·14-s + (1.46 + 0.843i)15-s + (0.211 − 0.366i)16-s + (−0.392 − 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.195229 + 0.0328838i\)
\(L(\frac12)\) \(\approx\) \(0.195229 + 0.0328838i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (867. + 8.28e3i)T \)
good2 \( 1 + (7.60 + 4.39i)T + (16 + 27.7i)T^{2} \)
3 \( 1 + (8.72 + 15.1i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (72.9 - 42.1i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (77.9 + 134. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 - 403.T + 1.61e5T^{2} \)
13 \( 1 + (257. - 148. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (468. + 270. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.05e3 + 606. i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 - 3.39e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.53e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.86e3iT - 2.86e7T^{2} \)
41 \( 1 + (-4.60e3 - 7.97e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 1.47e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.93e4T + 2.29e8T^{2} \)
53 \( 1 + (1.11e4 - 1.93e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (3.74e4 + 2.15e4i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (5.52e3 - 3.18e3i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.33e4 - 5.77e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (1.76e4 + 3.06e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + 3.29e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.03e4 + 1.17e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.85e4 - 4.95e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (2.77e4 + 1.60e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 3.50e4iT - 8.58e9T^{2} \)
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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

−15.86231702749665724420227227045, −14.09071307295064628022652626382, −12.34687313529843239621779872634, −11.57388841610207972396553268670, −10.69160986648362406042493421770, −9.222478552193559606866733117014, −7.37766279316395239649789229685, −7.03130993049568225367060301065, −3.49484483969786656113524200693, −1.08083242366733042378457988205, 0.23949427457711752782189203744, 4.27826986091832779585970681395, 6.07173057155560176283819342372, 7.82366399084157364476115465758, 8.965086366851591101038765474170, 9.850558251094936521692880013396, 11.38997749657683720655933808658, 12.46051344556804096615294807506, 15.13521329010029964138639582488, 15.64307808979612145343188043457

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