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Properties

Label 2-370-185.174-c1-0-8
Degree $2$
Conductor $370$
Sign $0.883 - 0.467i$
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  + (0.866 − 0.5i)2-s + (2.18 + 1.26i)3-s + (0.499 − 0.866i)4-s + (−0.837 + 2.07i)5-s + 2.52·6-s + (1.91 + 1.10i)7-s − 0.999i·8-s + (1.68 + 2.92i)9-s + (0.311 + 2.21i)10-s − 2.68·11-s + (2.18 − 1.26i)12-s + (−4.10 − 2.36i)13-s + 2.21·14-s + (−4.45 + 3.47i)15-s + (−0.5 − 0.866i)16-s + (−0.596 + 0.344i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (1.26 + 0.729i)3-s + (0.249 − 0.433i)4-s + (−0.374 + 0.927i)5-s + 1.03·6-s + (0.724 + 0.418i)7-s − 0.353i·8-s + (0.562 + 0.975i)9-s + (0.0983 + 0.700i)10-s − 0.810·11-s + (0.631 − 0.364i)12-s + (−1.13 − 0.657i)13-s + 0.591·14-s + (−1.14 + 0.897i)15-s + (−0.125 − 0.216i)16-s + (−0.144 + 0.0835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.883 - 0.467i$
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: \((2,\ 370,\ (\ :1/2),\ 0.883 - 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50890 + 0.622924i\)
\(L(\frac12)\) \(\approx\) \(2.50890 + 0.622924i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.837 - 2.07i)T \)
37 \( 1 + (6.08 + 0.178i)T \)
good3 \( 1 + (-2.18 - 1.26i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.91 - 1.10i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.68T + 11T^{2} \)
13 \( 1 + (4.10 + 2.36i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.596 - 0.344i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.59 + 4.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.21iT - 23T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 6.73T + 31T^{2} \)
41 \( 1 + (3.71 - 6.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 5.90iT - 43T^{2} \)
47 \( 1 - 4.83iT - 47T^{2} \)
53 \( 1 + (-0.680 + 0.392i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.130 - 0.225i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.73 - 4.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.31 + 1.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.90 - 8.49i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + (1.38 - 2.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.67 - 4.42i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.334 + 0.578i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−11.46886009960404288993297026247, −10.34188503927968822117783124368, −9.995007521522625648089860302144, −8.611932181631219706571579021204, −7.901030989079577351344162516552, −6.81724337781869520454790180001, −5.17751912117933947041953156314, −4.37736591357835487984589999061, −2.85438999017991369820390458598, −2.66855041330777398282365378323, 1.68990504768230671814377539079, 3.02579100916095526466224484315, 4.39566795045515537592095534307, 5.20433490361017577274780359967, 6.85472469876643174678290250877, 7.84209874088491147881099221110, 8.092058461295785056437736999018, 9.142383097348479062676764752140, 10.32964568310538357770611012347, 11.92836237746466647275634875857

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