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Properties

Label 2-37-37.10-c7-0-12
Degree $2$
Conductor $37$
Sign $0.0409 + 0.999i$
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  + (−2.85 + 4.94i)2-s + (−22.7 − 39.3i)3-s + (47.7 + 82.6i)4-s + (−48.1 − 83.4i)5-s + 259.·6-s + (330. + 572. i)7-s − 1.27e3·8-s + (62.6 − 108. i)9-s + 549.·10-s − 110.·11-s + (2.16e3 − 3.75e3i)12-s + (−5.92e3 − 1.02e4i)13-s − 3.77e3·14-s + (−2.18e3 + 3.78e3i)15-s + (−2.47e3 + 4.27e3i)16-s + (1.52e4 − 2.64e4i)17-s + ⋯
L(s)  = 1  + (−0.252 + 0.436i)2-s + (−0.485 − 0.840i)3-s + (0.372 + 0.645i)4-s + (−0.172 − 0.298i)5-s + 0.489·6-s + (0.364 + 0.630i)7-s − 0.880·8-s + (0.0286 − 0.0495i)9-s + 0.173·10-s − 0.0251·11-s + (0.361 − 0.626i)12-s + (−0.748 − 1.29i)13-s − 0.367·14-s + (−0.167 + 0.289i)15-s + (−0.150 + 0.261i)16-s + (0.753 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0409 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0409 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.0409 + 0.999i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.0409 + 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.693928 - 0.666038i\)
\(L(\frac12)\) \(\approx\) \(0.693928 - 0.666038i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (2.81e5 - 1.24e5i)T \)
good2 \( 1 + (2.85 - 4.94i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (22.7 + 39.3i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (48.1 + 83.4i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-330. - 572. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + 110.T + 1.94e7T^{2} \)
13 \( 1 + (5.92e3 + 1.02e4i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-1.52e4 + 2.64e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.11e4 + 3.66e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 - 6.60e4T + 3.40e9T^{2} \)
29 \( 1 + 2.19e4T + 1.72e10T^{2} \)
31 \( 1 + 2.20e5T + 2.75e10T^{2} \)
41 \( 1 + (-2.59e5 - 4.49e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 7.63e5T + 2.71e11T^{2} \)
47 \( 1 + 7.87e4T + 5.06e11T^{2} \)
53 \( 1 + (1.77e5 - 3.06e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-5.86e5 + 1.01e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.25e6 + 2.17e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-6.04e5 - 1.04e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-1.35e6 - 2.34e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 - 5.42e5T + 1.10e13T^{2} \)
79 \( 1 + (1.13e6 + 1.96e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (3.14e5 - 5.44e5i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (-4.59e5 + 7.95e5i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 4.10e6T + 8.07e13T^{2} \)
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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

−14.85747230325575361485884255616, −12.88501384747347327860182159714, −12.32077633328920680592582434715, −11.23229453607592253996816893001, −9.141846752738894225774001390518, −7.80098880881353227278860785639, −6.86151267222972431492322983891, −5.29744496439820711024242489302, −2.73550052152640643420785888928, −0.48788554055540211673626559134, 1.69223452300568406962677624078, 4.00722499373603300108452576851, 5.57719931206928727378124916953, 7.23797256652571238662311614781, 9.292625601481363134612862648677, 10.55911114022809149492911111292, 10.93750023737980797897054889375, 12.36101925044442625836351705126, 14.35875776375960731994981409190, 15.03757734707132164550977506709

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