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Properties

Label 2-37-37.6-c8-0-2
Degree $2$
Conductor $37$
Sign $0.526 + 0.850i$
Analytic cond. $15.0730$
Root an. cond. $3.88239$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−17.3 + 17.3i)2-s + 155. i·3-s − 346. i·4-s + (713. + 713. i)5-s + (−2.70e3 − 2.70e3i)6-s + 792.·7-s + (1.57e3 + 1.57e3i)8-s − 1.76e4·9-s − 2.47e4·10-s − 9.88e3i·11-s + 5.39e4·12-s + (−1.60e4 − 1.60e4i)13-s + (−1.37e4 + 1.37e4i)14-s + (−1.10e5 + 1.10e5i)15-s + 3.41e4·16-s + (−5.03e4 − 5.03e4i)17-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)2-s + 1.92i·3-s − 1.35i·4-s + (1.14 + 1.14i)5-s + (−2.08 − 2.08i)6-s + 0.329·7-s + (0.383 + 0.383i)8-s − 2.68·9-s − 2.47·10-s − 0.675i·11-s + 2.59·12-s + (−0.561 − 0.561i)13-s + (−0.357 + 0.357i)14-s + (−2.19 + 2.19i)15-s + 0.521·16-s + (−0.602 − 0.602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(15.0730\)
Root analytic conductor: \(3.88239\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :4),\ 0.526 + 0.850i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.545785 - 0.304149i\)
\(L(\frac12)\) \(\approx\) \(0.545785 - 0.304149i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.77e5 - 1.85e6i)T \)
good2 \( 1 + (17.3 - 17.3i)T - 256iT^{2} \)
3 \( 1 - 155. iT - 6.56e3T^{2} \)
5 \( 1 + (-713. - 713. i)T + 3.90e5iT^{2} \)
7 \( 1 - 792.T + 5.76e6T^{2} \)
11 \( 1 + 9.88e3iT - 2.14e8T^{2} \)
13 \( 1 + (1.60e4 + 1.60e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (5.03e4 + 5.03e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (-8.46e4 - 8.46e4i)T + 1.69e10iT^{2} \)
23 \( 1 + (2.42e5 + 2.42e5i)T + 7.83e10iT^{2} \)
29 \( 1 + (-1.45e5 + 1.45e5i)T - 5.00e11iT^{2} \)
31 \( 1 + (1.05e6 - 1.05e6i)T - 8.52e11iT^{2} \)
41 \( 1 - 3.72e6iT - 7.98e12T^{2} \)
43 \( 1 + (-2.13e6 - 2.13e6i)T + 1.16e13iT^{2} \)
47 \( 1 + 5.38e6T + 2.38e13T^{2} \)
53 \( 1 + 1.95e6T + 6.22e13T^{2} \)
59 \( 1 + (-3.34e6 - 3.34e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (-6.58e6 + 6.58e6i)T - 1.91e14iT^{2} \)
67 \( 1 + 1.00e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.49e7T + 6.45e14T^{2} \)
73 \( 1 - 3.40e7iT - 8.06e14T^{2} \)
79 \( 1 + (-1.54e7 - 1.54e7i)T + 1.51e15iT^{2} \)
83 \( 1 + 8.49e7T + 2.25e15T^{2} \)
89 \( 1 + (2.95e7 - 2.95e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-3.18e7 - 3.18e7i)T + 7.83e15iT^{2} \)
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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

−15.82343596693036642889404253661, −14.66306192973388685707301676807, −14.23303830651116589880096032686, −11.10158540865875322497411609971, −10.11861297843607655270892153581, −9.576485436685250752207069949030, −8.270966281083289901704493853104, −6.36710592626637034952978465902, −5.26622609996264551138800304405, −3.05063674143788953618004449222, 0.34177945377210183372219095098, 1.70567764785770227688400336741, 2.07344365073049011132380937882, 5.64926686064310178082382438942, 7.38836530842998315703145924675, 8.657198581348194730457146974851, 9.545662622133765785388290487981, 11.37189881220400757432890408826, 12.37793423898714267447064963547, 13.06603214592947327443161658682

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