↑

Properties

Label 2-370-37.9-c1-0-13
Degree $2$
Conductor $370$
Sign $-0.902 - 0.431i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.381 − 2.16i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s − 2.19·6-s + (−2.80 − 2.35i)7-s + (0.5 + 0.866i)8-s + (−1.72 − 0.626i)9-s + (−0.5 + 0.866i)10-s + (0.760 + 1.31i)11-s + (0.381 + 2.16i)12-s + (−0.396 + 0.144i)13-s + (−1.83 + 3.17i)14-s + (−1.68 + 1.41i)15-s + (0.766 − 0.642i)16-s + (−3.43 − 1.25i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.220 − 1.24i)3-s + (−0.469 + 0.171i)4-s + (−0.342 − 0.287i)5-s − 0.897·6-s + (−1.05 − 0.889i)7-s + (0.176 + 0.306i)8-s + (−0.574 − 0.208i)9-s + (−0.158 + 0.273i)10-s + (0.229 + 0.397i)11-s + (0.110 + 0.624i)12-s + (−0.110 + 0.0400i)13-s + (−0.489 + 0.847i)14-s + (−0.434 + 0.364i)15-s + (0.191 − 0.160i)16-s + (−0.833 − 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.902 - 0.431i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.902 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183490 + 0.808857i\)
\(L(\frac12)\) \(\approx\) \(0.183490 + 0.808857i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (4.81 + 3.71i)T \)
good3 \( 1 + (-0.381 + 2.16i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (2.80 + 2.35i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.760 - 1.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.396 - 0.144i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.43 + 1.25i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (0.280 - 1.59i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (-1.58 + 2.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.06 + 1.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.91T + 31T^{2} \)
41 \( 1 + (3.94 - 1.43i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 - 1.06T + 43T^{2} \)
47 \( 1 + (-1.17 + 2.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.03 + 1.70i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-10.3 + 8.69i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (11.3 - 4.12i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (9.69 + 8.13i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.80 + 15.9i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + (-1.70 - 1.42i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-12.2 - 4.44i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-2.42 + 2.03i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (7.19 - 12.4i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.92376872650071473002465399592, −10.00712223720142312236191525708, −9.059415163897745609749043681336, −8.014730383528656787313009500325, −7.10852182057677601905489031963, −6.42253587662930372478748681115, −4.62226331423091701365339514749, −3.46534953484958666911910428584, −2.07991287428108585350835972841, −0.56474996007858500611577681703, 2.95858892557149535146220761005, 3.94803570387815688987143927539, 5.09354920533794637988651324443, 6.17568703292291711904811595406, 7.08425873113930947595918030541, 8.563366297665164234547132062304, 9.082508997128773651481115841405, 9.896976258422144424803540826784, 10.72989875712620858517062063823, 11.81524337629551042909416945087

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