↑

Properties

Label 2-370-185.23-c1-0-4
Degree $2$
Conductor $370$
Sign $-0.0126 - 0.999i$
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  + (0.866 + 0.5i)2-s + (−2.83 − 0.758i)3-s + (0.499 + 0.866i)4-s + (−1.91 − 1.16i)5-s + (−2.07 − 2.07i)6-s + (2.40 + 0.644i)7-s + 0.999i·8-s + (4.84 + 2.79i)9-s + (−1.07 − 1.96i)10-s + 2.91i·11-s + (−0.758 − 2.83i)12-s + (−2.54 + 1.47i)13-s + (1.76 + 1.76i)14-s + (4.53 + 4.73i)15-s + (−0.5 + 0.866i)16-s + (−2.25 + 3.89i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.63 − 0.438i)3-s + (0.249 + 0.433i)4-s + (−0.854 − 0.518i)5-s + (−0.846 − 0.846i)6-s + (0.909 + 0.243i)7-s + 0.353i·8-s + (1.61 + 0.932i)9-s + (−0.340 − 0.619i)10-s + 0.878i·11-s + (−0.219 − 0.817i)12-s + (−0.706 + 0.407i)13-s + (0.470 + 0.470i)14-s + (1.17 + 1.22i)15-s + (−0.125 + 0.216i)16-s + (−0.545 + 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0126 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0126 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.0126 - 0.999i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.0126 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591331 + 0.598884i\)
\(L(\frac12)\) \(\approx\) \(0.591331 + 0.598884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (1.91 + 1.16i)T \)
37 \( 1 + (5.53 + 2.51i)T \)
good3 \( 1 + (2.83 + 0.758i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-2.40 - 0.644i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 - 2.91iT - 11T^{2} \)
13 \( 1 + (2.54 - 1.47i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.25 - 3.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.54 - 0.949i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.99iT - 23T^{2} \)
29 \( 1 + (-3.87 - 3.87i)T + 29iT^{2} \)
31 \( 1 + (-3.48 + 3.48i)T - 31iT^{2} \)
41 \( 1 + (4.86 - 2.81i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.87iT - 43T^{2} \)
47 \( 1 + (-6.90 + 6.90i)T - 47iT^{2} \)
53 \( 1 + (9.56 - 2.56i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.43 - 5.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.95 - 1.32i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.901 - 3.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.12 - 14.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.91 - 1.91i)T - 73iT^{2} \)
79 \( 1 + (4.93 + 1.32i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (9.27 - 2.48i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (5.84 - 1.56i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + 1.22T + 97T^{2} \)
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

−11.79761906867292394290614795305, −11.24196773385791024332824363126, −10.09539262554667786643442131979, −8.557320748721915834299138342092, −7.44697383761389853763482593287, −6.92741399300898700101471194724, −5.53519750641329171507309745761, −4.98847505248608064476290691701, −4.08729300353729802799024719379, −1.63588329524304691184460085878, 0.60158473974736866851833754022, 3.04193732655950193468268250663, 4.59661617493827347589151797128, 4.89038060713901952023247000636, 6.19930440876893380435223389813, 7.03637612362686480728213129143, 8.214450194582396736602525545782, 9.861342467095446045927449749767, 10.78808367462859863215618302286, 11.17464377053694444467113680233

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