Properties

Label 2-380-19.11-c1-0-5
Degree $2$
Conductor $380$
Sign $0.447 + 0.894i$
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  + (1.26 − 2.18i)3-s + (−0.5 + 0.866i)5-s + 2.72·7-s + (−1.67 − 2.90i)9-s + 3.31·11-s + (−1.62 − 2.81i)13-s + (1.26 + 2.18i)15-s + (1.17 − 2.03i)17-s + (−3.11 + 3.04i)19-s + (3.43 − 5.95i)21-s + (−1.07 − 1.86i)23-s + (−0.499 − 0.866i)25-s − 0.893·27-s + (1.96 + 3.40i)29-s − 10.1·31-s + ⋯
L(s)  = 1  + (0.727 − 1.26i)3-s + (−0.223 + 0.387i)5-s + 1.03·7-s + (−0.559 − 0.968i)9-s + 0.999·11-s + (−0.450 − 0.780i)13-s + (0.325 + 0.563i)15-s + (0.285 − 0.494i)17-s + (−0.714 + 0.699i)19-s + (0.750 − 1.29i)21-s + (−0.223 − 0.387i)23-s + (−0.0999 − 0.173i)25-s − 0.171·27-s + (0.365 + 0.632i)29-s − 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54095 - 0.952012i\)
\(L(\frac12)\) \(\approx\) \(1.54095 - 0.952012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (3.11 - 3.04i)T \)
good3 \( 1 + (-1.26 + 2.18i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.72T + 7T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
13 \( 1 + (1.62 + 2.81i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.17 + 2.03i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.07 + 1.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.96 - 3.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 3.68T + 37T^{2} \)
41 \( 1 + (-0.363 + 0.629i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.18 + 2.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.51 - 9.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.49 - 7.77i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.48 + 9.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.22 - 7.32i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.87 - 8.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.45 - 5.99i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.24 + 2.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.99 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 + (4.27 + 7.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.61 - 6.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.34243626168811471846588523610, −10.39106828550345252261150883214, −9.034699477021568230936051014680, −8.227332066461852114191053030253, −7.49180250503373422809292603406, −6.77631177304371483819394490817, −5.48707127908677488243485917875, −3.96582561127663072115051724034, −2.58104200211940855445101081780, −1.39795602157506891481188536356, 1.96908588031302520275192266506, 3.73248915548215278393516785151, 4.36485583685468051003612318099, 5.32233138334260588054563674682, 6.90400435701199192527706506818, 8.144675283000637328687940240142, 8.895520703592685193183815762988, 9.489747012537142359488760140733, 10.54438836845327964768214834849, 11.40363297963930455595563687520

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