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Properties

Label 2-37-37.26-c7-0-16
Degree $2$
Conductor $37$
Sign $-0.752 - 0.658i$
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  + (−8.43 − 14.6i)2-s + (−26.6 + 46.1i)3-s + (−78.4 + 135. i)4-s + (196. − 340. i)5-s + 900.·6-s + (448. − 776. i)7-s + 486.·8-s + (−328. − 568. i)9-s − 6.63e3·10-s − 2.37e3·11-s + (−4.18e3 − 7.24e3i)12-s + (1.29e3 − 2.24e3i)13-s − 1.51e4·14-s + (1.04e4 + 1.81e4i)15-s + (5.93e3 + 1.02e4i)16-s + (−1.07e4 − 1.87e4i)17-s + ⋯
L(s)  = 1  + (−0.745 − 1.29i)2-s + (−0.570 + 0.987i)3-s + (−0.612 + 1.06i)4-s + (0.703 − 1.21i)5-s + 1.70·6-s + (0.494 − 0.855i)7-s + 0.336·8-s + (−0.150 − 0.260i)9-s − 2.09·10-s − 0.539·11-s + (−0.698 − 1.21i)12-s + (0.163 − 0.283i)13-s − 1.47·14-s + (0.801 + 1.38i)15-s + (0.361 + 0.626i)16-s + (−0.533 − 0.923i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ -0.752 - 0.658i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.139882 + 0.371939i\)
\(L(\frac12)\) \(\approx\) \(0.139882 + 0.371939i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.74e5 - 1.40e5i)T \)
good2 \( 1 + (8.43 + 14.6i)T + (-64 + 110. i)T^{2} \)
3 \( 1 + (26.6 - 46.1i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-196. + 340. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-448. + 776. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 2.37e3T + 1.94e7T^{2} \)
13 \( 1 + (-1.29e3 + 2.24e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (1.07e4 + 1.87e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (1.57e4 - 2.72e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + 7.28e4T + 3.40e9T^{2} \)
29 \( 1 + 1.36e5T + 1.72e10T^{2} \)
31 \( 1 - 6.30e4T + 2.75e10T^{2} \)
41 \( 1 + (-8.66e4 + 1.50e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + 5.90e5T + 2.71e11T^{2} \)
47 \( 1 - 1.07e6T + 5.06e11T^{2} \)
53 \( 1 + (2.40e5 + 4.17e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (1.21e6 + 2.10e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.16e6 + 2.01e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-3.92e3 + 6.79e3i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (2.66e6 - 4.61e6i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 - 2.59e6T + 1.10e13T^{2} \)
79 \( 1 + (2.90e6 - 5.02e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-3.94e6 - 6.82e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (4.21e6 + 7.30e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 3.85e6T + 8.07e13T^{2} \)
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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

−13.73615804610743404730635341098, −12.55922866234553863706545464364, −11.25486644921532527158912309200, −10.30912567019557849279349790120, −9.583761753328478440474367864028, −8.247772685240959570702632735879, −5.41961884324307109120613428730, −4.11872035517685054284038685871, −1.75541715680659252510348544563, −0.23278244329652474758796658524, 2.11259388581287696637236833867, 5.78294178821223591196063598825, 6.47159765327819674575833912481, 7.53610606981176446287734264539, 8.908477618656904361278734871954, 10.50766725431971193550023156383, 11.92677438522359775387795416540, 13.45011036411162141624335887612, 14.76895243637886253839844839551, 15.50325303031295731003590479881

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