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Properties

Label 2-37-37.11-c5-0-5
Degree $2$
Conductor $37$
Sign $-0.0125 - 0.999i$
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  + (−9.18 + 5.30i)2-s + (−6.26 + 10.8i)3-s + (40.2 − 69.7i)4-s + (63.9 + 36.9i)5-s − 132. i·6-s + (113. − 196. i)7-s + 514. i·8-s + (43.0 + 74.5i)9-s − 783.·10-s + 134.·11-s + (504. + 873. i)12-s + (−145. − 84.0i)13-s + 2.40e3i·14-s + (−801. + 462. i)15-s + (−1.44e3 − 2.49e3i)16-s + (444. − 256. i)17-s + ⋯
L(s)  = 1  + (−1.62 + 0.937i)2-s + (−0.401 + 0.695i)3-s + (1.25 − 2.17i)4-s + (1.14 + 0.660i)5-s − 1.50i·6-s + (0.873 − 1.51i)7-s + 2.84i·8-s + (0.177 + 0.306i)9-s − 2.47·10-s + 0.335·11-s + (1.01 + 1.75i)12-s + (−0.238 − 0.137i)13-s + 3.27i·14-s + (−0.919 + 0.530i)15-s + (−1.40 − 2.43i)16-s + (0.372 − 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0125 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0125 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.0125 - 0.999i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.0125 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.619266 + 0.627070i\)
\(L(\frac12)\) \(\approx\) \(0.619266 + 0.627070i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-8.08e3 + 1.99e3i)T \)
good2 \( 1 + (9.18 - 5.30i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (6.26 - 10.8i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-63.9 - 36.9i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-113. + 196. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 134.T + 1.61e5T^{2} \)
13 \( 1 + (145. + 84.0i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-444. + 256. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.94e3 - 1.12e3i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 2.76e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.02e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.03e3iT - 2.86e7T^{2} \)
41 \( 1 + (3.29e3 - 5.70e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 6.09e3iT - 1.47e8T^{2} \)
47 \( 1 - 7.38e3T + 2.29e8T^{2} \)
53 \( 1 + (6.00e3 + 1.04e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.31e4 + 1.33e4i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.92e4 + 1.68e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.01e3 - 3.49e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.01e4 + 1.75e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + 1.13e4T + 2.07e9T^{2} \)
79 \( 1 + (-2.97e4 - 1.71e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-4.52e4 - 7.83e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (2.72e4 - 1.57e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 5.36e4iT - 8.58e9T^{2} \)
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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

−16.32234272316479206054866303204, −14.61867043030995161453860132187, −14.04350686389290501574106998563, −10.98081257419867722123218597681, −10.37199731883439722950087562904, −9.628931282669784605742101165759, −7.83322307009337854097323333749, −6.76334993653047251087005188343, −5.20283219076777099000984221765, −1.34761232518231710062361828052, 1.15513126102944802383619004536, 2.22255688070132265009620831515, 5.81675470231880523636328402874, 7.68738234186277262205260740657, 9.088786123280708836174186749567, 9.658672412093381782003315104748, 11.60675070030405693496787762236, 12.01229831004486443717776887237, 13.29666407880505133156552898658, 15.45569594743481432784628905928

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