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Properties

Label 2-37-37.27-c3-0-7
Degree $2$
Conductor $37$
Sign $-0.488 - 0.872i$
Analytic cond. $2.18307$
Root an. cond. $1.47752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.61 − 2.08i)2-s + (−3.72 − 6.45i)3-s + (4.70 + 8.15i)4-s + (−2.02 + 1.16i)5-s + 31.1i·6-s + (3.33 + 5.77i)7-s − 5.90i·8-s + (−14.3 + 24.7i)9-s + 9.76·10-s − 62.1·11-s + (35.1 − 60.8i)12-s + (43.8 − 25.3i)13-s − 27.8i·14-s + (15.1 + 8.72i)15-s + (25.3 − 43.8i)16-s + (−37.8 − 21.8i)17-s + ⋯
L(s)  = 1  + (−1.27 − 0.737i)2-s + (−0.717 − 1.24i)3-s + (0.588 + 1.01i)4-s + (−0.181 + 0.104i)5-s + 2.11i·6-s + (0.179 + 0.311i)7-s − 0.260i·8-s + (−0.530 + 0.918i)9-s + 0.308·10-s − 1.70·11-s + (0.844 − 1.46i)12-s + (0.935 − 0.540i)13-s − 0.531i·14-s + (0.260 + 0.150i)15-s + (0.395 − 0.685i)16-s + (−0.540 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ -0.488 - 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0803545 + 0.137095i\)
\(L(\frac12)\) \(\approx\) \(0.0803545 + 0.137095i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-216. + 62.4i)T \)
good2 \( 1 + (3.61 + 2.08i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (3.72 + 6.45i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (2.02 - 1.16i)T + (62.5 - 108. i)T^{2} \)
7 \( 1 + (-3.33 - 5.77i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + 62.1T + 1.33e3T^{2} \)
13 \( 1 + (-43.8 + 25.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (37.8 + 21.8i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (129. - 74.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 71.8iT - 1.21e4T^{2} \)
29 \( 1 + 136. iT - 2.43e4T^{2} \)
31 \( 1 + 10.8iT - 2.97e4T^{2} \)
41 \( 1 + (19.0 + 32.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 452. iT - 7.95e4T^{2} \)
47 \( 1 - 116.T + 1.03e5T^{2} \)
53 \( 1 + (45.3 - 78.5i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (476. + 275. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (379. - 218. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (177. + 308. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-522. - 904. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 790.T + 3.89e5T^{2} \)
79 \( 1 + (-176. + 101. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-327. + 567. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (79.8 + 46.1i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 307. iT - 9.12e5T^{2} \)
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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

−15.41256361812474688679170848591, −13.34439120771779239360397801762, −12.40848436890979750112404502125, −11.20596379828587677521560167942, −10.41861989483089759454520206316, −8.497275275255705604090968775941, −7.60770437135783476148710424133, −5.83207742858346389161084367815, −2.16823903942751996899734313237, −0.21298552752718506213037874656, 4.48603180814409695098554820666, 6.20800178116343622192036483269, 7.947460450682345445428151436492, 9.152628035020442640521461905725, 10.49074949066490597314927289752, 10.98890278251317749814153422344, 13.14444944427401473096175597952, 15.21121000496790519826262727934, 15.83560951852199344520202397488, 16.57319561234820942533358201475

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