↑

Properties

Label 2-370-185.174-c1-0-11
Degree $2$
Conductor $370$
Sign $0.986 + 0.162i$
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 + (1.41 + 0.815i)3-s + (0.499 − 0.866i)4-s + (2.14 + 0.618i)5-s + 1.63·6-s + (−0.466 − 0.269i)7-s − 0.999i·8-s + (−0.170 − 0.294i)9-s + (2.17 − 0.539i)10-s − 0.829·11-s + (1.41 − 0.815i)12-s + (−0.945 − 0.545i)13-s − 0.539·14-s + (2.53 + 2.62i)15-s + (−0.5 − 0.866i)16-s + (1.01 − 0.585i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.815 + 0.470i)3-s + (0.249 − 0.433i)4-s + (0.961 + 0.276i)5-s + 0.665·6-s + (−0.176 − 0.101i)7-s − 0.353i·8-s + (−0.0566 − 0.0981i)9-s + (0.686 − 0.170i)10-s − 0.250·11-s + (0.407 − 0.235i)12-s + (−0.262 − 0.151i)13-s − 0.144·14-s + (0.653 + 0.677i)15-s + (−0.125 − 0.216i)16-s + (0.245 − 0.141i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.986 + 0.162i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.986 + 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55535 - 0.209489i\)
\(L(\frac12)\) \(\approx\) \(2.55535 - 0.209489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.14 - 0.618i)T \)
37 \( 1 + (-4.29 + 4.30i)T \)
good3 \( 1 + (-1.41 - 0.815i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.466 + 0.269i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.829T + 11T^{2} \)
13 \( 1 + (0.945 + 0.545i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.01 + 0.585i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.87 - 6.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.539iT - 23T^{2} \)
29 \( 1 + 9.57T + 29T^{2} \)
31 \( 1 - 3.09T + 31T^{2} \)
41 \( 1 + (0.960 - 1.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.29iT - 43T^{2} \)
47 \( 1 - 5.80iT - 47T^{2} \)
53 \( 1 + (-3.06 + 1.76i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.95 - 3.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.908 + 1.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.0 + 6.39i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.290 - 0.503i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.75iT - 73T^{2} \)
79 \( 1 + (4.65 - 8.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.86 + 1.07i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.39 + 9.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.17iT - 97T^{2} \)
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

−11.31125464043077024498938648004, −10.24414752034137288930832863010, −9.778712347858516895492511355252, −8.842885957550237548482697321899, −7.64845989050594571949432321053, −6.29886242258848461423397262550, −5.52728034949395419483585037175, −4.10676307667860299908768728579, −3.12299092614910901649427770538, −2.00762210868214363487569047403, 2.04070613049101012015052358194, 2.95016686189519207428471442057, 4.58435168223422397078829686595, 5.60433504187068422651312666665, 6.63939794608235170186425590176, 7.58611509227847646415879537139, 8.627839493741335117329746185302, 9.310787814919293078447563404470, 10.50510831761873143129098790282, 11.56447898433998703796543385450

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