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Properties

Label 2-37-37.11-c7-0-11
Degree $2$
Conductor $37$
Sign $0.824 - 0.565i$
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  + (3.63 − 2.09i)2-s + (24.6 − 42.7i)3-s + (−55.2 + 95.6i)4-s + (292. + 169. i)5-s − 206. i·6-s + (−480. + 831. i)7-s + 1.00e3i·8-s + (−122. − 211. i)9-s + 1.41e3·10-s + 333.·11-s + (2.72e3 + 4.71e3i)12-s + (4.18e3 + 2.41e3i)13-s + 4.02e3i·14-s + (1.44e4 − 8.33e3i)15-s + (−4.96e3 − 8.60e3i)16-s + (1.58e4 − 9.12e3i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.185i)2-s + (0.527 − 0.913i)3-s + (−0.431 + 0.746i)4-s + (1.04 + 0.604i)5-s − 0.390i·6-s + (−0.529 + 0.916i)7-s + 0.690i·8-s + (−0.0559 − 0.0968i)9-s + 0.448·10-s + 0.0754·11-s + (0.454 + 0.787i)12-s + (0.528 + 0.305i)13-s + 0.392i·14-s + (1.10 − 0.637i)15-s + (−0.303 − 0.525i)16-s + (0.780 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.824 - 0.565i$
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: \((2,\ 37,\ (\ :7/2),\ 0.824 - 0.565i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.38802 + 0.739852i\)
\(L(\frac12)\) \(\approx\) \(2.38802 + 0.739852i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.27e5 + 2.07e5i)T \)
good2 \( 1 + (-3.63 + 2.09i)T + (64 - 110. i)T^{2} \)
3 \( 1 + (-24.6 + 42.7i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-292. - 169. i)T + (3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (480. - 831. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 - 333.T + 1.94e7T^{2} \)
13 \( 1 + (-4.18e3 - 2.41e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-1.58e4 + 9.12e3i)T + (2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-7.84e3 - 4.53e3i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 5.74e4iT - 3.40e9T^{2} \)
29 \( 1 - 5.61e4iT - 1.72e10T^{2} \)
31 \( 1 + 1.35e5iT - 2.75e10T^{2} \)
41 \( 1 + (2.16e5 - 3.74e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + 6.67e5iT - 2.71e11T^{2} \)
47 \( 1 + 5.69e5T + 5.06e11T^{2} \)
53 \( 1 + (2.35e5 + 4.07e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (-2.23e6 + 1.28e6i)T + (1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.76e6 - 1.01e6i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (6.14e5 - 1.06e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (1.51e6 - 2.63e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 - 2.37e6T + 1.10e13T^{2} \)
79 \( 1 + (5.10e6 + 2.94e6i)T + (9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (1.48e6 + 2.56e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (-7.69e6 + 4.44e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 2.55e6iT - 8.07e13T^{2} \)
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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

−14.41940718225389620748595421414, −13.61932615073041034051904801963, −12.85585115844285582637668005952, −11.70867582159141346710490430763, −9.762015199300981114465069483681, −8.545812648768580035349701889469, −7.13110925147077695537409921669, −5.62362318914789415043000799332, −3.19186458169138277548842244665, −2.01877187981511164502899976461, 1.05895378996842997594832500670, 3.68352142150055392947817271199, 5.03059214380328350797472752711, 6.41904398379600481521871453658, 8.770413888799510697937207338127, 9.872082361901520313950411650337, 10.36120106954369736708892824233, 12.83702031557598535427997503854, 13.74861679747266305831328287294, 14.59165855476456189370398028695

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