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Properties

Label 2-37-37.27-c7-0-11
Degree $2$
Conductor $37$
Sign $0.174 - 0.984i$
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  + (14.2 + 8.23i)2-s + (23.0 + 39.9i)3-s + (71.7 + 124. i)4-s + (363. − 209. i)5-s + 759. i·6-s + (415. + 720. i)7-s + 255. i·8-s + (32.0 − 55.4i)9-s + 6.91e3·10-s − 276.·11-s + (−3.30e3 + 5.72e3i)12-s + (−1.29e4 + 7.49e3i)13-s + 1.37e4i·14-s + (1.67e4 + 9.66e3i)15-s + (7.08e3 − 1.22e4i)16-s + (−3.33e4 − 1.92e4i)17-s + ⋯
L(s)  = 1  + (1.26 + 0.728i)2-s + (0.492 + 0.853i)3-s + (0.560 + 0.970i)4-s + (1.29 − 0.750i)5-s + 1.43i·6-s + (0.458 + 0.793i)7-s + 0.176i·8-s + (0.0146 − 0.0253i)9-s + 2.18·10-s − 0.0626·11-s + (−0.552 + 0.956i)12-s + (−1.63 + 0.945i)13-s + 1.33i·14-s + (1.28 + 0.739i)15-s + (0.432 − 0.748i)16-s + (−1.64 − 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.174 - 0.984i$
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: \((2,\ 37,\ (\ :7/2),\ 0.174 - 0.984i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.51126 + 2.94431i\)
\(L(\frac12)\) \(\approx\) \(3.51126 + 2.94431i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-2.83e5 - 1.21e5i)T \)
good2 \( 1 + (-14.2 - 8.23i)T + (64 + 110. i)T^{2} \)
3 \( 1 + (-23.0 - 39.9i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-363. + 209. i)T + (3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-415. - 720. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + 276.T + 1.94e7T^{2} \)
13 \( 1 + (1.29e4 - 7.49e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (3.33e4 + 1.92e4i)T + (2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-1.02e4 + 5.93e3i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 - 7.82e3iT - 3.40e9T^{2} \)
29 \( 1 - 1.18e5iT - 1.72e10T^{2} \)
31 \( 1 + 7.10e4iT - 2.75e10T^{2} \)
41 \( 1 + (-1.04e4 - 1.81e4i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + 7.90e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.06e6T + 5.06e11T^{2} \)
53 \( 1 + (1.80e5 - 3.11e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-9.53e4 - 5.50e4i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (1.00e6 - 5.80e5i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.23e6 - 2.14e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (2.62e5 + 4.53e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 2.78e6T + 1.10e13T^{2} \)
79 \( 1 + (3.60e6 - 2.07e6i)T + (9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (6.42e5 - 1.11e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (-8.79e6 - 5.07e6i)T + (2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 9.69e6iT - 8.07e13T^{2} \)
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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

−14.93049923545266785987312368831, −14.13769245597979714393051843629, −13.13758782702741837713017534400, −11.93055521665932563087919417218, −9.695173988175370488120345763407, −9.043827502399494722198850207581, −6.79353078177035674918198885905, −5.21776111143805998587394187325, −4.57996772186973480397001874892, −2.41503169212207085982594246063, 1.84594784218618715818177797860, 2.70972523386955984329191673224, 4.75029978627854333828036657451, 6.36569851110759847341251335228, 7.80957377189523437983119299693, 10.06423585724363113378701358342, 11.02520820752970997167982207114, 12.72893096897141803328949424583, 13.33349899854428255256768052681, 14.19662171475992858168510332939

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