Properties

Label 2-380-19.5-c1-0-5
Degree $2$
Conductor $380$
Sign $-0.849 + 0.527i$
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  + (0.543 − 3.08i)3-s + (−0.766 − 0.642i)5-s + (0.0481 + 0.0833i)7-s + (−6.37 − 2.32i)9-s + (0.190 − 0.329i)11-s + (−0.668 − 3.79i)13-s + (−2.39 + 2.01i)15-s + (2.49 − 0.909i)17-s + (−0.788 + 4.28i)19-s + (0.283 − 0.103i)21-s + (0.131 − 0.110i)23-s + (0.173 + 0.984i)25-s + (−5.92 + 10.2i)27-s + (−3.67 − 1.33i)29-s + (−1.23 − 2.13i)31-s + ⋯
L(s)  = 1  + (0.313 − 1.77i)3-s + (−0.342 − 0.287i)5-s + (0.0181 + 0.0315i)7-s + (−2.12 − 0.773i)9-s + (0.0573 − 0.0994i)11-s + (−0.185 − 1.05i)13-s + (−0.618 + 0.519i)15-s + (0.605 − 0.220i)17-s + (−0.180 + 0.983i)19-s + (0.0617 − 0.0224i)21-s + (0.0273 − 0.0229i)23-s + (0.0347 + 0.196i)25-s + (−1.14 + 1.97i)27-s + (−0.682 − 0.248i)29-s + (−0.221 − 0.383i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.849 + 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336855 - 1.18222i\)
\(L(\frac12)\) \(\approx\) \(0.336855 - 1.18222i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.788 - 4.28i)T \)
good3 \( 1 + (-0.543 + 3.08i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-0.0481 - 0.0833i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.190 + 0.329i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.668 + 3.79i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-2.49 + 0.909i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.131 + 0.110i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.67 + 1.33i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.23 + 2.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + (-0.688 + 3.90i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-9.81 - 8.23i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (10.5 + 3.84i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-9.54 + 8.00i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-5.79 + 2.10i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.93 - 3.29i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-11.4 - 4.16i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.30 - 1.09i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.258 + 1.46i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.60 + 14.7i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.46 + 9.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.400 + 2.27i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.37 + 0.500i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.37097410144899956234132381674, −10.02417760585727650981445102649, −8.780489971601836749591880506044, −7.891978735408903071778627132677, −7.50686420506403422311708102424, −6.28225384825922054079542016160, −5.43509843503513700113854328340, −3.56093221549864961408204670418, −2.26912502090536573161364769167, −0.811045728740192393632785599695, 2.67843427951203647009597744529, 3.86920687271814072908267068209, 4.56866012931917474245087520321, 5.67987904475250700972176126559, 7.07942691082371814052565666666, 8.310672643408859551500710352140, 9.251356994718058513256970562404, 9.761342718364732059926155837284, 10.89208962977883610917943113645, 11.26040888284745147038267090852

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