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Properties

Label 2-37-37.25-c1-0-2
Degree $2$
Conductor $37$
Sign $-0.684 + 0.729i$
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  + (−1.61 − 1.93i)2-s + (−0.968 − 0.812i)3-s + (−0.755 + 4.28i)4-s + (−0.681 − 1.87i)5-s + 3.18i·6-s + (2.99 − 1.09i)7-s + (5.12 − 2.95i)8-s + (−0.243 − 1.37i)9-s + (−2.51 + 4.34i)10-s + (2.73 + 4.73i)11-s + (4.21 − 3.53i)12-s + (−1.50 − 0.265i)13-s + (−6.96 − 4.02i)14-s + (−0.861 + 2.36i)15-s + (−5.83 − 2.12i)16-s + (−0.794 + 0.140i)17-s + ⋯
L(s)  = 1  + (−1.14 − 1.36i)2-s + (−0.559 − 0.469i)3-s + (−0.377 + 2.14i)4-s + (−0.304 − 0.837i)5-s + 1.30i·6-s + (1.13 − 0.412i)7-s + (1.81 − 1.04i)8-s + (−0.0810 − 0.459i)9-s + (−0.793 + 1.37i)10-s + (0.823 + 1.42i)11-s + (1.21 − 1.02i)12-s + (−0.418 − 0.0737i)13-s + (−1.86 − 1.07i)14-s + (−0.222 + 0.611i)15-s + (−1.45 − 0.531i)16-s + (−0.192 + 0.0339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.684 + 0.729i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ -0.684 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160826 - 0.371413i\)
\(L(\frac12)\) \(\approx\) \(0.160826 - 0.371413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-0.539 - 6.05i)T \)
good2 \( 1 + (1.61 + 1.93i)T + (-0.347 + 1.96i)T^{2} \)
3 \( 1 + (0.968 + 0.812i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (0.681 + 1.87i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-2.99 + 1.09i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.50 + 0.265i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.794 - 0.140i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (1.35 - 1.61i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (-3.02 - 1.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.122 - 0.0705i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.37iT - 31T^{2} \)
41 \( 1 + (-0.0732 + 0.415i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 2.00iT - 43T^{2} \)
47 \( 1 + (0.842 - 1.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.86 + 2.49i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (3.65 - 10.0i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-11.3 - 2.00i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.25 - 2.64i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-6.78 - 5.69i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 8.77T + 73T^{2} \)
79 \( 1 + (-1.54 - 4.24i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.81 + 10.3i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-0.914 + 2.51i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (13.3 + 7.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−16.94750967270518053690542039030, −14.83825580404394750630970812312, −12.81854180837438138289770082550, −12.05016229135350104976255301596, −11.31383848195747382949118533325, −9.843403202470334991287704989345, −8.649498873925465138918540952458, −7.33404626355660138137539384454, −4.41994963930206506080330484970, −1.42671988979252199490618197658, 5.16074017364710416220264506083, 6.52379962657937814436237699976, 7.939559817984124993724705087374, 9.021021894851236055673586099684, 10.71933793343202522631695451034, 11.32870510151204364818682699321, 14.15963954207834712390884127389, 14.85011316067061366069553674822, 15.96621579467481279036200870064, 16.87532385889704888757354362706

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