Properties

Label 2-370-37.10-c1-0-3
Degree $2$
Conductor $370$
Sign $0.729 - 0.683i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (1.5 + 2.59i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 3·6-s − 0.999·8-s + (−3 + 5.19i)9-s + 0.999·10-s + 2·11-s + (1.50 − 2.59i)12-s + (0.5 + 0.866i)13-s + (−1.5 + 2.59i)15-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (3 + 5.19i)18-s + (−1 − 1.73i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 + 1.49i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + 1.22·6-s − 0.353·8-s + (−1 + 1.73i)9-s + 0.316·10-s + 0.603·11-s + (0.433 − 0.749i)12-s + (0.138 + 0.240i)13-s + (−0.387 + 0.670i)15-s + (−0.125 + 0.216i)16-s + (0.242 − 0.420i)17-s + (0.707 + 1.22i)18-s + (−0.229 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.729 - 0.683i$
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: \((2,\ 370,\ (\ :1/2),\ 0.729 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93869 + 0.766476i\)
\(L(\frac12)\) \(\approx\) \(1.93869 + 0.766476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.5 + 2.59i)T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.32321448255447855739248646614, −10.38904104770267075409135561590, −9.857731872583700502391303430650, −9.045073460082600879588754395068, −8.188868981264419101140333349812, −6.58063465243093294159014663586, −5.23746815707710114612601892896, −4.23807683064305712673795393728, −3.42054882189283528330028759162, −2.30393138872681614184910660274, 1.40283922902562266868125513243, 2.84862013964375338544521413445, 4.21201875262721930949205950758, 5.88171599656515378543321409274, 6.52228201100071469565806930147, 7.57527665202680956664320514258, 8.299301297591984611693436244611, 8.946564042132285791414141588271, 10.19166817942503132232718048092, 11.95163200795000568405637685859

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