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Properties

Label 2-380-19.9-c1-0-2
Degree $2$
Conductor $380$
Sign $0.904 - 0.426i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
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  + (−2.38 + 1.99i)3-s + (0.939 + 0.342i)5-s + (2.42 − 4.19i)7-s + (1.15 − 6.55i)9-s + (−0.912 − 1.57i)11-s + (4.37 + 3.67i)13-s + (−2.91 + 1.06i)15-s + (0.843 + 4.78i)17-s + (3.11 − 3.04i)19-s + (2.61 + 14.8i)21-s + (−3.49 + 1.27i)23-s + (0.766 + 0.642i)25-s + (5.67 + 9.83i)27-s + (0.509 − 2.88i)29-s + (−0.598 + 1.03i)31-s + ⋯
L(s)  = 1  + (−1.37 + 1.15i)3-s + (0.420 + 0.152i)5-s + (0.915 − 1.58i)7-s + (0.385 − 2.18i)9-s + (−0.275 − 0.476i)11-s + (1.21 + 1.01i)13-s + (−0.753 + 0.274i)15-s + (0.204 + 1.16i)17-s + (0.715 − 0.698i)19-s + (0.570 + 3.23i)21-s + (−0.728 + 0.264i)23-s + (0.153 + 0.128i)25-s + (1.09 + 1.89i)27-s + (0.0945 − 0.536i)29-s + (−0.107 + 0.186i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.904 - 0.426i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03357 + 0.231347i\)
\(L(\frac12)\) \(\approx\) \(1.03357 + 0.231347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-3.11 + 3.04i)T \)
good3 \( 1 + (2.38 - 1.99i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-2.42 + 4.19i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.912 + 1.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.37 - 3.67i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.843 - 4.78i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (3.49 - 1.27i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.509 + 2.88i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (0.598 - 1.03i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.79T + 37T^{2} \)
41 \( 1 + (-6.35 + 5.33i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-9.07 - 3.30i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.728 + 4.12i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (1.62 - 0.590i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.96 - 11.1i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.03 + 0.377i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.781 - 4.43i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (6.95 + 2.53i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (5.48 - 4.60i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-2.27 + 1.90i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-5.31 + 9.20i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.63 - 1.37i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.46 + 8.30i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.09177422970743790775679918854, −10.71698273796814326738912858039, −9.980638563810413139358736182926, −8.887883863077098615001608425155, −7.53360482306108254343067350432, −6.35157908341489345120051325477, −5.60229958346992836299760511786, −4.35690901188200451279198574283, −3.87467113004121336860686159177, −1.12117292142370320334710521878, 1.25054334959559957789252632942, 2.48471248123406263251746602988, 4.95318115746292254221754387691, 5.65384085946100672880480675844, 6.15442054851177742452176463731, 7.57595327515206972914889500530, 8.219684641069664931413429674260, 9.481214474950452303174196163941, 10.77829633268014740790740902780, 11.43327811372038136305097453001

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