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Properties

Label 2-370-37.9-c1-0-9
Degree $2$
Conductor $370$
Sign $-0.625 + 0.780i$
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  + (−0.173 − 0.984i)2-s + (−0.483 + 2.74i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + 2.78·6-s + (−2.74 − 2.30i)7-s + (0.5 + 0.866i)8-s + (−4.46 − 1.62i)9-s + (−0.5 + 0.866i)10-s + (−1.18 − 2.04i)11-s + (−0.483 − 2.74i)12-s + (1.42 − 0.517i)13-s + (−1.78 + 3.09i)14-s + (2.13 − 1.79i)15-s + (0.766 − 0.642i)16-s + (−0.105 − 0.0385i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.279 + 1.58i)3-s + (−0.469 + 0.171i)4-s + (−0.342 − 0.287i)5-s + 1.13·6-s + (−1.03 − 0.869i)7-s + (0.176 + 0.306i)8-s + (−1.48 − 0.542i)9-s + (−0.158 + 0.273i)10-s + (−0.356 − 0.617i)11-s + (−0.139 − 0.791i)12-s + (0.394 − 0.143i)13-s + (−0.478 + 0.828i)14-s + (0.550 − 0.462i)15-s + (0.191 − 0.160i)16-s + (−0.0256 − 0.00934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.625 + 0.780i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161422 - 0.336117i\)
\(L(\frac12)\) \(\approx\) \(0.161422 - 0.336117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (4.70 - 3.84i)T \)
good3 \( 1 + (0.483 - 2.74i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (2.74 + 2.30i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (1.18 + 2.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.42 + 0.517i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.105 + 0.0385i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-1.05 + 6.00i)T + (-17.8 - 6.49i)T^{2} \)
23 \( 1 + (-0.0857 + 0.148i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.37 + 5.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.78T + 31T^{2} \)
41 \( 1 + (-11.4 + 4.16i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + 6.91T + 43T^{2} \)
47 \( 1 + (4.30 - 7.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.934 - 0.784i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.587 - 0.493i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.72 - 0.990i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.0466 - 0.0391i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.17 - 6.64i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + 1.90T + 73T^{2} \)
79 \( 1 + (9.61 + 8.07i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.48 - 0.539i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-12.1 + 10.1i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-8.37 + 14.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.96179409481615016775624121050, −10.25064346271986351248207580076, −9.433347954931178760000660672062, −8.815778303825451307135003377803, −7.42042301435516343634342955722, −5.91543405672748240878972683630, −4.78524496260879360959925528898, −3.84805681021265790891119440154, −3.13101474233333310616312947410, −0.26357529248656297142596775960, 1.87994208629219650115309860183, 3.45987477387736097905314305068, 5.45401916405332721901048118907, 6.16786443853997572611829569085, 7.03462320986081646676141815046, 7.67583948665514687779465194010, 8.681312995566176639316012732537, 9.673885761129837753942766808390, 10.94512227379062173586985364622, 12.11574462514026365325427759321

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