Properties

Label 2-370-37.10-c1-0-4
Degree $2$
Conductor $370$
Sign $0.639 + 0.768i$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + (0.581 + 1.00i)7-s − 0.999·8-s + (1 − 1.73i)9-s − 0.999·10-s + 5.16·11-s + (0.499 − 0.866i)12-s + (−1.08 − 1.87i)13-s + 1.16·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (2.58 − 4.47i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.408·6-s + (0.219 + 0.380i)7-s − 0.353·8-s + (0.333 − 0.577i)9-s − 0.316·10-s + 1.55·11-s + (0.144 − 0.249i)12-s + (−0.299 − 0.519i)13-s + 0.310·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.626 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.639 + 0.768i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.639 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63579 - 0.767094i\)
\(L(\frac12)\) \(\approx\) \(1.63579 - 0.767094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (2.91 - 5.33i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.581 - 1.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.16T + 11T^{2} \)
13 \( 1 + (1.08 + 1.87i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.58 + 4.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.16 - 3.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 + 4.32T + 29T^{2} \)
31 \( 1 + 2.16T + 31T^{2} \)
41 \( 1 + (-3.66 - 6.34i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.32T + 43T^{2} \)
47 \( 1 + 1.16T + 47T^{2} \)
53 \( 1 + (3.91 - 6.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.58 - 4.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.16 + 3.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.16 + 3.74i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.16 - 5.47i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.32 - 7.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.324T + 97T^{2} \)
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

−11.67704083956038108705936766812, −10.19349415627479744058213059125, −9.519837969353871172753268222413, −8.858881987252620074431055414914, −7.60976537116963888834486477735, −6.27181354985175491471744186825, −5.12210213656408051973019799627, −4.05048485559214208465038423359, −3.20139536268897994387666769347, −1.35447221325611620025392050118, 1.78310007598689425396275651776, 3.57777546859040679066802749497, 4.49224624422850542773502210331, 5.91201945061375624048425417175, 6.98858934388703801375299692271, 7.47585057635906747897134028797, 8.535021558184385101449312019398, 9.513920048940083851637040144679, 10.70979411503800669922476043141, 11.69104901758846195621931441462

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