Properties

Label 2-37-37.7-c1-0-0
Degree $2$
Conductor $37$
Sign $0.612 - 0.790i$
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  + (−1.76 + 0.642i)2-s + (1.26 + 0.460i)3-s + (1.17 − 0.984i)4-s + (0.532 + 3.01i)5-s − 2.53·6-s + (−0.613 − 3.47i)7-s + (0.439 − 0.761i)8-s + (−0.907 − 0.761i)9-s + (−2.87 − 4.98i)10-s + (0.173 − 0.300i)11-s + (1.93 − 0.705i)12-s + (−0.141 + 0.118i)13-s + (3.31 + 5.74i)14-s + (−0.716 + 4.06i)15-s + (−0.819 + 4.64i)16-s + (−4.44 − 3.72i)17-s + ⋯
L(s)  = 1  + (−1.24 + 0.454i)2-s + (0.730 + 0.266i)3-s + (0.586 − 0.492i)4-s + (0.237 + 1.34i)5-s − 1.03·6-s + (−0.231 − 1.31i)7-s + (0.155 − 0.269i)8-s + (−0.302 − 0.253i)9-s + (−0.910 − 1.57i)10-s + (0.0523 − 0.0906i)11-s + (0.559 − 0.203i)12-s + (−0.0392 + 0.0329i)13-s + (0.887 + 1.53i)14-s + (−0.185 + 1.04i)15-s + (−0.204 + 1.16i)16-s + (−1.07 − 0.904i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468344 + 0.229497i\)
\(L(\frac12)\) \(\approx\) \(0.468344 + 0.229497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.52 - 5.88i)T \)
good2 \( 1 + (1.76 - 0.642i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-1.26 - 0.460i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.532 - 3.01i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.613 + 3.47i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.173 + 0.300i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.141 - 0.118i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (4.44 + 3.72i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-6.31 - 2.29i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-0.266 - 0.460i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.03 - 6.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.75T + 31T^{2} \)
41 \( 1 + (-1.70 + 1.43i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 1.14T + 43T^{2} \)
47 \( 1 + (-0.620 - 1.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0675 + 0.383i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.724 + 4.10i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (7.07 - 5.93i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.354 + 2.01i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-12.7 - 4.63i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + (1.32 + 7.50i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.401 + 0.337i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-1.75 + 9.95i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-1.53 - 2.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.75718683871369503665869146789, −15.62060099268559665729670154887, −14.36341140035885086900025837373, −13.58709238600789001389426491331, −11.12958177829900258256392927737, −10.09420580141117476721610475108, −9.166580197699551051080048460220, −7.57038307099854411842851544129, −6.75306636850893663883797849058, −3.44230437478349435697206350255, 2.16312392434262654231502416857, 5.37966362007576547207566923392, 7.973233751276848054085129141488, 8.940923193338128235128359532195, 9.389584956066054845544265941393, 11.30187519460758053021493048913, 12.58087181553799953054911576464, 13.71920481029575251594235622984, 15.35068977203353526264247306999, 16.58780304213291158003743679921

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