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Properties

Label 2-37-37.27-c1-0-0
Degree $2$
Conductor $37$
Sign $0.989 - 0.146i$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (1.36 + 2.36i)3-s + (−0.500 − 0.866i)4-s + (0.232 − 0.133i)5-s − 2.73i·6-s + (−1.73 − 3i)7-s + 3i·8-s + (−2.23 + 3.86i)9-s − 0.267·10-s − 4.73·11-s + (1.36 − 2.36i)12-s + (3 − 1.73i)13-s + 3.46i·14-s + (0.633 + 0.366i)15-s + (0.500 − 0.866i)16-s + (3.23 + 1.86i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.788 + 1.36i)3-s + (−0.250 − 0.433i)4-s + (0.103 − 0.0599i)5-s − 1.11i·6-s + (−0.654 − 1.13i)7-s + 1.06i·8-s + (−0.744 + 1.28i)9-s − 0.0847·10-s − 1.42·11-s + (0.394 − 0.683i)12-s + (0.832 − 0.480i)13-s + 0.925i·14-s + (0.163 + 0.0945i)15-s + (0.125 − 0.216i)16-s + (0.783 + 0.452i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635782 + 0.0468470i\)
\(L(\frac12)\) \(\approx\) \(0.635782 + 0.0468470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-0.5 + 6.06i)T \)
good2 \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.36 - 2.36i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.232 + 0.133i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.73 + 3i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
13 \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.23 - 1.86i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.732iT - 23T^{2} \)
29 \( 1 + 0.267iT - 29T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 2.19T + 47T^{2} \)
53 \( 1 + (-1.26 + 2.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.6 + 6.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.69 + 2.13i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.63 - 6.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (-3.29 + 1.90i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.09 + 7.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.96 - 4.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.26iT - 97T^{2} \)
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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

−16.29536175559085193918651236273, −15.44792655887832267679264244076, −14.19564210271264530672186267789, −13.25541289459997628579891953039, −10.66542876398739700939841692566, −10.34099957354795295676075951142, −9.266411750997812188935422250362, −7.981710621679176466758122142059, −5.27240718439870873687442968740, −3.47142450387139421332160195020, 2.83419209981600773873845599856, 6.21368134975019881540540966060, 7.67398301586931020584067523025, 8.463055515120429309058165103598, 9.667180646149637157853060340803, 12.04491838717221638246372863411, 12.97726896420630993775678932614, 13.72975969604880907608878017365, 15.41706229786082009184797924641, 16.45155250800410297356032232738

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