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Properties

Label 2-37-37.27-c7-0-2
Degree $2$
Conductor $37$
Sign $0.830 + 0.556i$
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  + (−13.4 − 7.78i)2-s + (−38.3 − 66.4i)3-s + (57.0 + 98.8i)4-s + (4.26 − 2.46i)5-s + 1.19e3i·6-s + (243. + 420. i)7-s + 215. i·8-s + (−1.84e3 + 3.20e3i)9-s − 76.6·10-s + 1.53e3·11-s + (4.37e3 − 7.58e3i)12-s + (−4.90e3 + 2.82e3i)13-s − 7.56e3i·14-s + (−327. − 188. i)15-s + (8.98e3 − 1.55e4i)16-s + (8.03e3 + 4.63e3i)17-s + ⋯
L(s)  = 1  + (−1.19 − 0.687i)2-s + (−0.820 − 1.42i)3-s + (0.445 + 0.772i)4-s + (0.0152 − 0.00881i)5-s + 2.25i·6-s + (0.267 + 0.463i)7-s + 0.149i·8-s + (−0.844 + 1.46i)9-s − 0.0242·10-s + 0.346·11-s + (0.731 − 1.26i)12-s + (−0.618 + 0.357i)13-s − 0.736i·14-s + (−0.0250 − 0.0144i)15-s + (0.548 − 0.949i)16-s + (0.396 + 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.830 + 0.556i$
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: \((2,\ 37,\ (\ :7/2),\ 0.830 + 0.556i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.467881 - 0.142230i\)
\(L(\frac12)\) \(\approx\) \(0.467881 - 0.142230i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.25e5 - 2.10e5i)T \)
good2 \( 1 + (13.4 + 7.78i)T + (64 + 110. i)T^{2} \)
3 \( 1 + (38.3 + 66.4i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-4.26 + 2.46i)T + (3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-243. - 420. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 1.53e3T + 1.94e7T^{2} \)
13 \( 1 + (4.90e3 - 2.82e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + (-8.03e3 - 4.63e3i)T + (2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-2.77e3 + 1.60e3i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + 2.93e3iT - 3.40e9T^{2} \)
29 \( 1 - 7.27e4iT - 1.72e10T^{2} \)
31 \( 1 + 2.37e5iT - 2.75e10T^{2} \)
41 \( 1 + (-3.82e5 - 6.63e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 3.24e5iT - 2.71e11T^{2} \)
47 \( 1 - 4.38e5T + 5.06e11T^{2} \)
53 \( 1 + (-7.05e5 + 1.22e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-2.56e6 - 1.48e6i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-2.11e6 + 1.22e6i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-3.82e5 - 6.62e5i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-5.48e5 - 9.50e5i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 4.35e6T + 1.10e13T^{2} \)
79 \( 1 + (1.17e6 - 6.79e5i)T + (9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (4.52e6 - 7.83e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (9.39e3 + 5.42e3i)T + (2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 9.61e6iT - 8.07e13T^{2} \)
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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

−14.60395010348856858949064013041, −13.06921156714787428538721630396, −11.86157472970588596945740870966, −11.34942329184697020753719008407, −9.766947030771892796207534306045, −8.312351027205621057267880306183, −7.15357792084246923515794292628, −5.56741265498593037823651890674, −2.18967122468534577346185940189, −1.02276435445169592467879482750, 0.46110694017662476857492871786, 4.04592229856477485735033366590, 5.62743308883055904978825231037, 7.24984125473397885785156693397, 8.830876049449243404269390639182, 9.979121738987019196632514017612, 10.63771889054012195344328117818, 12.12583325078940946696001343689, 14.33382324698407382044751611103, 15.60160989190835879584035387210

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