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Properties

Label 2-37-37.36-c3-0-5
Degree $2$
Conductor $37$
Sign $0.586 + 0.810i$
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  − 2.49i·2-s + 5.45·3-s + 1.77·4-s + 5.37i·5-s − 13.6i·6-s − 8.18·7-s − 24.3i·8-s + 2.73·9-s + 13.4·10-s − 16.1·11-s + 9.69·12-s + 25.4i·13-s + 20.4i·14-s + 29.3i·15-s − 46.6·16-s + 42.9i·17-s + ⋯
L(s)  = 1  − 0.881i·2-s + 1.04·3-s + 0.222·4-s + 0.481i·5-s − 0.925i·6-s − 0.442·7-s − 1.07i·8-s + 0.101·9-s + 0.424·10-s − 0.443·11-s + 0.233·12-s + 0.543i·13-s + 0.389i·14-s + 0.504i·15-s − 0.728·16-s + 0.612i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.586 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.586 + 0.810i$
Analytic conductor: \(2.18307\)
Root analytic conductor: \(1.47752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :3/2),\ 0.586 + 0.810i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.53396 - 0.783323i\)
\(L(\frac12)\) \(\approx\) \(1.53396 - 0.783323i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-131. - 182. i)T \)
good2 \( 1 + 2.49iT - 8T^{2} \)
3 \( 1 - 5.45T + 27T^{2} \)
5 \( 1 - 5.37iT - 125T^{2} \)
7 \( 1 + 8.18T + 343T^{2} \)
11 \( 1 + 16.1T + 1.33e3T^{2} \)
13 \( 1 - 25.4iT - 2.19e3T^{2} \)
17 \( 1 - 42.9iT - 4.91e3T^{2} \)
19 \( 1 - 90.2iT - 6.85e3T^{2} \)
23 \( 1 + 91.4iT - 1.21e4T^{2} \)
29 \( 1 + 233. iT - 2.43e4T^{2} \)
31 \( 1 - 81.9iT - 2.97e4T^{2} \)
41 \( 1 + 197.T + 6.89e4T^{2} \)
43 \( 1 + 258. iT - 7.95e4T^{2} \)
47 \( 1 - 202.T + 1.03e5T^{2} \)
53 \( 1 - 142.T + 1.48e5T^{2} \)
59 \( 1 + 610. iT - 2.05e5T^{2} \)
61 \( 1 - 184. iT - 2.26e5T^{2} \)
67 \( 1 - 414.T + 3.00e5T^{2} \)
71 \( 1 - 1.16e3T + 3.57e5T^{2} \)
73 \( 1 - 316.T + 3.89e5T^{2} \)
79 \( 1 + 1.24e3iT - 4.93e5T^{2} \)
83 \( 1 + 277.T + 5.71e5T^{2} \)
89 \( 1 - 829. iT - 7.04e5T^{2} \)
97 \( 1 - 1.76e3iT - 9.12e5T^{2} \)
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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

−15.48668331492306859853430018781, −14.45626044472735421597036876739, −13.26361284861524431769040154757, −12.07125285298476245255799798120, −10.70237935746880204035360433639, −9.669495216375025623431674674024, −8.148052847447817328128292582001, −6.52997025386277423716920602153, −3.62327125316605440196122914665, −2.34157561726656982925440354486, 2.84209136377387485282669692008, 5.33200618577179860286216396053, 7.08788874556800578417784776865, 8.255401901337357283641152580535, 9.322330803895851319446733355098, 11.15462344205755586929810737475, 12.85946109896502221787393598232, 13.96644717064251278143061540126, 15.06086110723412249053839429754, 15.87288595968083068383101297654

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