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Properties

Label 2-37-37.27-c1-0-1
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 + (−0.366 − 0.633i)3-s + (−0.500 − 0.866i)4-s + (−3.23 + 1.86i)5-s − 0.732i·6-s + (1.73 + 3i)7-s − 3i·8-s + (1.23 − 2.13i)9-s − 3.73·10-s − 1.26·11-s + (−0.366 + 0.633i)12-s + (3 − 1.73i)13-s + 3.46i·14-s + (2.36 + 1.36i)15-s + (0.500 − 0.866i)16-s + (−0.232 − 0.133i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.211 − 0.366i)3-s + (−0.250 − 0.433i)4-s + (−1.44 + 0.834i)5-s − 0.298i·6-s + (0.654 + 1.13i)7-s − 1.06i·8-s + (0.410 − 0.711i)9-s − 1.18·10-s − 0.382·11-s + (−0.105 + 0.183i)12-s + (0.832 − 0.480i)13-s + 0.925i·14-s + (0.610 + 0.352i)15-s + (0.125 − 0.216i)16-s + (−0.0562 − 0.0324i)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

\(L(1)\) \(\approx\) \(0.798313 + 0.0588229i\)
\(L(\frac12)\) \(\approx\) \(0.798313 + 0.0588229i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.366 + 0.633i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.23 - 1.86i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.73 - 3i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.232 + 0.133i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.73iT - 23T^{2} \)
29 \( 1 - 3.73iT - 29T^{2} \)
31 \( 1 - 2.19iT - 31T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 8.19T + 47T^{2} \)
53 \( 1 + (-4.73 + 8.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.66 - 3.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.69 - 3.86i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.36 - 9.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + (12.2 - 7.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.09 - 1.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.03 - 0.598i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.73iT - 97T^{2} \)
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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

−15.77553039652723882649511900436, −15.27825243588373152454344077786, −14.49463384722823721759206426631, −12.82142626557256032763234170957, −11.83719433424265678692159522193, −10.61099437881020159914628383184, −8.615746368426324840805977490267, −7.11439599612220304666639934573, −5.71895029011267292383692650485, −3.85722758301771213994367792375, 4.13590772308870469558933499340, 4.63135584885078132368899514754, 7.63988918284055115668979239788, 8.503065513922239613514066855566, 10.79307589990860853553547145219, 11.56789388667199248067801007828, 12.87799197537873568434232376727, 13.74998711924527348158039136717, 15.33935438521367381583522697989, 16.47612963263808562418766425119

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