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Properties

Label 2-380-19.16-c1-0-4
Degree $2$
Conductor $380$
Sign $0.247 + 0.968i$
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  + (1.02 − 0.374i)3-s + (0.173 − 0.984i)5-s + (−1.81 − 3.14i)7-s + (−1.38 + 1.15i)9-s + (2.71 − 4.71i)11-s + (1.31 + 0.479i)13-s + (−0.189 − 1.07i)15-s + (−2.12 − 1.78i)17-s + (1.94 − 3.90i)19-s + (−3.04 − 2.55i)21-s + (1.15 + 6.52i)23-s + (−0.939 − 0.342i)25-s + (−2.62 + 4.55i)27-s + (4.29 − 3.60i)29-s + (5.34 + 9.26i)31-s + ⋯
L(s)  = 1  + (0.593 − 0.216i)3-s + (0.0776 − 0.440i)5-s + (−0.685 − 1.18i)7-s + (−0.460 + 0.386i)9-s + (0.819 − 1.42i)11-s + (0.365 + 0.133i)13-s + (−0.0490 − 0.278i)15-s + (−0.515 − 0.432i)17-s + (0.446 − 0.894i)19-s + (−0.663 − 0.556i)21-s + (0.240 + 1.36i)23-s + (−0.187 − 0.0684i)25-s + (−0.505 + 0.875i)27-s + (0.797 − 0.669i)29-s + (0.960 + 1.66i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.247 + 0.968i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.247 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17647 - 0.913407i\)
\(L(\frac12)\) \(\approx\) \(1.17647 - 0.913407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-1.94 + 3.90i)T \)
good3 \( 1 + (-1.02 + 0.374i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.81 + 3.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.71 + 4.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.31 - 0.479i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.12 + 1.78i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.15 - 6.52i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-4.29 + 3.60i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-5.34 - 9.26i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.37T + 37T^{2} \)
41 \( 1 + (2.35 - 0.858i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.84 + 10.4i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.67 + 3.08i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-1.60 - 9.10i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.79 - 1.50i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.509 + 2.88i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.64 - 1.38i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.83 - 16.1i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (12.1 - 4.41i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-10.7 + 3.89i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.51 - 9.55i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-10.5 - 3.83i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (7.22 + 6.06i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.17361555884252817111380010127, −10.25611662822233701347785984123, −9.039926214753669252078949383137, −8.619036849684220184092152031892, −7.38675115711731291992043759027, −6.54799249791915463774490601933, −5.30983572795188231207874808573, −3.88618989435476854450752328054, −2.97408573831773308444665238688, −0.996814919982386616523436531897, 2.20985224191035459686939047281, 3.23659604415404865938628923050, 4.45952093454893687719766915337, 6.04843783283732027333920805771, 6.58480375697928347960301715296, 8.009964407409746964904467866434, 8.959749593631127515986972744932, 9.557716511006312014678022790473, 10.42099366295937415111065078154, 11.80565978957934135949692493219

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