↑

Properties

Label 2-370-185.84-c1-0-11
Degree $2$
Conductor $370$
Sign $0.726 + 0.687i$
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 + (0.499 + 0.866i)4-s + (1.86 − 1.23i)5-s + (2.36 − 1.36i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (−2.23 + 0.133i)10-s + 1.26·11-s + (1.26 − 0.732i)13-s − 2.73·14-s + (−0.5 + 0.866i)16-s + (−1.5 − 0.866i)17-s + (2.59 − 1.5i)18-s + (0.633 + 1.09i)19-s + (1.99 + i)20-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.834 − 0.550i)5-s + (0.894 − 0.516i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.705 + 0.0423i)10-s + 0.382·11-s + (0.351 − 0.203i)13-s − 0.730·14-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.210i)17-s + (0.612 − 0.353i)18-s + (0.145 + 0.251i)19-s + (0.447 + 0.223i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16639 - 0.464422i\)
\(L(\frac12)\) \(\approx\) \(1.16639 - 0.464422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 1.23i)T \)
37 \( 1 + (-2.59 + 5.5i)T \)
good3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.36 + 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
13 \( 1 + (-1.26 + 0.732i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.633 - 1.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 1.73T + 29T^{2} \)
31 \( 1 - 7.66T + 31T^{2} \)
41 \( 1 + (2.96 + 5.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 + (7.26 + 4.19i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.73 - 8.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.86 + 6.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.56 - 4.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.56 - 7.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + (-7.83 - 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.19 + 1.26i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.23 - 5.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.19iT - 97T^{2} \)
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

−11.09416465511206506578246617618, −10.43053470668519026322906156047, −9.476951032888920985791892732497, −8.477792462110918306809766267965, −7.913450999327177245951913710069, −6.57529689787088997706179133007, −5.35247627093593017980828345601, −4.33458277560117063201350108713, −2.53040825982049553068896613635, −1.28483020143125582087777913855, 1.57424150231961967945257086400, 3.00117636607862617056858553965, 4.81457003188789232725764168001, 6.07104127152737529247015358375, 6.55367934146860195117975878771, 7.895672064782375657627623634292, 8.855829530424660130989055771263, 9.471516056542884577463229214632, 10.52578924543267199495534534921, 11.40665974668540626601567987354

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