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Properties

Label 2-37-37.31-c2-0-0
Degree $2$
Conductor $37$
Sign $0.491 - 0.871i$
Analytic cond. $1.00817$
Root an. cond. $1.00408$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.15 − 2.15i)2-s + 5.58i·3-s + 5.31i·4-s + (−2.97 + 2.97i)5-s + (12.0 − 12.0i)6-s + 8.27·7-s + (2.82 − 2.82i)8-s − 22.1·9-s + 12.8·10-s + 6.70i·11-s − 29.6·12-s + (4.81 − 4.81i)13-s + (−17.8 − 17.8i)14-s + (−16.6 − 16.6i)15-s + 9.04·16-s + (8.97 − 8.97i)17-s + ⋯
L(s)  = 1  + (−1.07 − 1.07i)2-s + 1.86i·3-s + 1.32i·4-s + (−0.595 + 0.595i)5-s + (2.00 − 2.00i)6-s + 1.18·7-s + (0.353 − 0.353i)8-s − 2.46·9-s + 1.28·10-s + 0.609i·11-s − 2.47·12-s + (0.370 − 0.370i)13-s + (−1.27 − 1.27i)14-s + (−1.10 − 1.10i)15-s + 0.565·16-s + (0.527 − 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.491 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.491 - 0.871i$
Analytic conductor: \(1.00817\)
Root analytic conductor: \(1.00408\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1),\ 0.491 - 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.508387 + 0.296996i\)
\(L(\frac12)\) \(\approx\) \(0.508387 + 0.296996i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-6.96 - 36.3i)T \)
good2 \( 1 + (2.15 + 2.15i)T + 4iT^{2} \)
3 \( 1 - 5.58iT - 9T^{2} \)
5 \( 1 + (2.97 - 2.97i)T - 25iT^{2} \)
7 \( 1 - 8.27T + 49T^{2} \)
11 \( 1 - 6.70iT - 121T^{2} \)
13 \( 1 + (-4.81 + 4.81i)T - 169iT^{2} \)
17 \( 1 + (-8.97 + 8.97i)T - 289iT^{2} \)
19 \( 1 + (2.58 - 2.58i)T - 361iT^{2} \)
23 \( 1 + (-22.5 + 22.5i)T - 529iT^{2} \)
29 \( 1 + (-22.9 - 22.9i)T + 841iT^{2} \)
31 \( 1 + (7.78 + 7.78i)T + 961iT^{2} \)
41 \( 1 - 2.70iT - 1.68e3T^{2} \)
43 \( 1 + (0.467 - 0.467i)T - 1.84e3iT^{2} \)
47 \( 1 + 4.93T + 2.20e3T^{2} \)
53 \( 1 - 21.4T + 2.80e3T^{2} \)
59 \( 1 + (-15.2 + 15.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (53.6 + 53.6i)T + 3.72e3iT^{2} \)
67 \( 1 + 64.0iT - 4.48e3T^{2} \)
71 \( 1 + 79.3T + 5.04e3T^{2} \)
73 \( 1 + 80.3iT - 5.32e3T^{2} \)
79 \( 1 + (-30.6 + 30.6i)T - 6.24e3iT^{2} \)
83 \( 1 - 144.T + 6.88e3T^{2} \)
89 \( 1 + (105. + 105. i)T + 7.92e3iT^{2} \)
97 \( 1 + (57.6 - 57.6i)T - 9.40e3iT^{2} \)
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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

−16.55841431560053919549444810074, −15.16989906495697668749432448591, −14.56571932660251649786284795499, −11.88430881738641644282672564154, −10.97214423673393883634545090805, −10.41870211528043926311207328160, −9.183674319137143323997670551484, −8.072145275021823872756750426286, −4.85106509947345654662408430694, −3.18289942557877272407931282772, 1.10160782993406000975875803797, 5.78938299904279924002106091267, 7.20010634592662375133240653854, 8.117627018995941042407857184235, 8.704756667660568945679499277045, 11.26573351026308687833770438360, 12.35387427112304636341358264111, 13.73839141613133358907202707428, 14.88160079971831916315443402172, 16.39036362650764635109035303579

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