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Properties

Label 2-37-37.11-c7-0-13
Degree $2$
Conductor $37$
Sign $0.980 - 0.197i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.1 − 7.04i)2-s + (−13.3 + 23.1i)3-s + (35.2 − 60.9i)4-s + (158. + 91.2i)5-s + 376. i·6-s + (380. − 659. i)7-s + 811. i·8-s + (736. + 1.27e3i)9-s + 2.57e3·10-s + 783.·11-s + (940. + 1.62e3i)12-s + (1.36e4 + 7.86e3i)13-s − 1.07e4i·14-s + (−4.22e3 + 2.43e3i)15-s + (1.02e4 + 1.77e4i)16-s + (8.35e3 − 4.82e3i)17-s + ⋯
L(s)  = 1  + (1.07 − 0.622i)2-s + (−0.285 + 0.494i)3-s + (0.275 − 0.476i)4-s + (0.565 + 0.326i)5-s + 0.711i·6-s + (0.419 − 0.727i)7-s + 0.560i·8-s + (0.336 + 0.583i)9-s + 0.813·10-s + 0.177·11-s + (0.157 + 0.272i)12-s + (1.71 + 0.992i)13-s − 1.04i·14-s + (−0.323 + 0.186i)15-s + (0.623 + 1.08i)16-s + (0.412 − 0.238i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.980 - 0.197i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.980 - 0.197i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.22967 + 0.322634i\)
\(L(\frac12)\) \(\approx\) \(3.22967 + 0.322634i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.28e5 + 2.80e5i)T \)
good2 \( 1 + (-12.1 + 7.04i)T + (64 - 110. i)T^{2} \)
3 \( 1 + (13.3 - 23.1i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-158. - 91.2i)T + (3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-380. + 659. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 - 783.T + 1.94e7T^{2} \)
13 \( 1 + (-1.36e4 - 7.86e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-8.35e3 + 4.82e3i)T + (2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (4.23e4 + 2.44e4i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + 7.12e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.64e5iT - 1.72e10T^{2} \)
31 \( 1 - 2.49e5iT - 2.75e10T^{2} \)
41 \( 1 + (-2.02e5 + 3.51e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + 3.61e5iT - 2.71e11T^{2} \)
47 \( 1 + 9.28e5T + 5.06e11T^{2} \)
53 \( 1 + (2.35e5 + 4.07e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-1.35e6 + 7.84e5i)T + (1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.72e6 + 9.97e5i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.46e6 + 2.53e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-9.79e5 + 1.69e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 - 3.47e6T + 1.10e13T^{2} \)
79 \( 1 + (-8.27e5 - 4.78e5i)T + (9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (9.55e5 + 1.65e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (8.62e6 - 4.98e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.09e7iT - 8.07e13T^{2} \)
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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

−14.32085366035688699339503220773, −13.79600346534087742078274115477, −12.63042445855417521842054401992, −10.98140007168175556207016828003, −10.66605022403445887816567449939, −8.638255025673587234857509241729, −6.53435088224529133297040495817, −4.88637344024958032383358170993, −3.84055241387109148516913409520, −1.87903870729936907924464484914, 1.31742662446927398179844349464, 3.85198171683480323517705844955, 5.71902013464578479082562375503, 6.18580246315270706108210959022, 8.093147859471617572569692747216, 9.748545986393828070888847349526, 11.59790807298516233522165047751, 12.90736147078837650689949600810, 13.38713321011777917370881035170, 14.92098477262994584182236789137

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