Properties

Label 2-37-37.27-c7-0-15
Degree $2$
Conductor $37$
Sign $0.645 + 0.764i$
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  + (15.6 + 9.06i)2-s + (−28.3 − 49.0i)3-s + (100. + 173. i)4-s + (169. − 98.0i)5-s − 1.02e3i·6-s + (−758. − 1.31e3i)7-s + 1.31e3i·8-s + (−509. + 881. i)9-s + 3.55e3·10-s + 5.03e3·11-s + (5.67e3 − 9.83e3i)12-s + (−4.12e3 + 2.38e3i)13-s − 2.74e4i·14-s + (−9.61e3 − 5.55e3i)15-s + (918. − 1.59e3i)16-s + (−2.72e3 − 1.57e3i)17-s + ⋯
L(s)  = 1  + (1.38 + 0.801i)2-s + (−0.605 − 1.04i)3-s + (0.783 + 1.35i)4-s + (0.607 − 0.350i)5-s − 1.93i·6-s + (−0.835 − 1.44i)7-s + 0.907i·8-s + (−0.232 + 0.403i)9-s + 1.12·10-s + 1.14·11-s + (0.948 − 1.64i)12-s + (−0.520 + 0.300i)13-s − 2.67i·14-s + (−0.735 − 0.424i)15-s + (0.0560 − 0.0970i)16-s + (−0.134 − 0.0777i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.645 + 0.764i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.645 + 0.764i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.79714 - 1.29902i\)
\(L(\frac12)\) \(\approx\) \(2.79714 - 1.29902i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-15.6 - 9.06i)T + (64 + 110. i)T^{2} \)
3 \( 1 + (28.3 + 49.0i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-169. + 98.0i)T + (3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (758. + 1.31e3i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 5.03e3T + 1.94e7T^{2} \)
13 \( 1 + (4.12e3 - 2.38e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (2.72e3 + 1.57e3i)T + (2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-3.75e4 + 2.17e4i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 - 2.30e4iT - 3.40e9T^{2} \)
29 \( 1 - 2.95e4iT - 1.72e10T^{2} \)
31 \( 1 - 2.01e5iT - 2.75e10T^{2} \)
41 \( 1 + (1.50e5 + 2.61e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 2.77e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.15e6T + 5.06e11T^{2} \)
53 \( 1 + (-7.61e5 + 1.31e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (1.22e5 + 7.06e4i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-2.44e6 + 1.41e6i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.36e6 - 2.36e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-1.93e6 - 3.34e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 3.25e6T + 1.10e13T^{2} \)
79 \( 1 + (5.07e6 - 2.92e6i)T + (9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (-2.15e6 + 3.73e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (2.35e6 + 1.35e6i)T + (2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 1.25e7iT - 8.07e13T^{2} \)
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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

−14.15026262048355501692786783451, −13.57218462946405831456200169543, −12.72224762834640955830064453258, −11.68915059915396870656194846929, −9.664855278856371086427048310274, −7.04868798829100258783073170631, −6.84334065555412884019470411906, −5.37269616797780184219156384701, −3.72832632695602340741500694432, −1.02182810082261173526448395097, 2.38367360777969318849115258828, 3.82118214569225718968583093069, 5.37545963418662835390807631507, 6.11598645829095297642746623716, 9.380751425953950952360201955589, 10.29315140559126022552120485676, 11.68818428442687645955127556667, 12.34280910870731993376131815085, 13.79179825890749447271859869357, 14.90956384611877730283195611286

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