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Properties

Label 2-380-19.17-c1-0-1
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: $\chi_{380} (321, \cdot )$
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

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

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