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Properties

Label 2-380-76.27-c1-0-19
Degree $2$
Conductor $380$
Sign $0.417 + 0.908i$
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  + (−0.104 − 1.41i)2-s + (−0.481 + 0.834i)3-s + (−1.97 + 0.293i)4-s + (0.5 − 0.866i)5-s + (1.22 + 0.592i)6-s + 1.00i·7-s + (0.620 + 2.75i)8-s + (1.03 + 1.79i)9-s + (−1.27 − 0.615i)10-s − 4.53i·11-s + (0.708 − 1.79i)12-s + (4.62 − 2.66i)13-s + (1.42 − 0.105i)14-s + (0.481 + 0.834i)15-s + (3.82 − 1.16i)16-s + (2.17 − 3.76i)17-s + ⋯
L(s)  = 1  + (−0.0736 − 0.997i)2-s + (−0.278 + 0.481i)3-s + (−0.989 + 0.146i)4-s + (0.223 − 0.387i)5-s + (0.501 + 0.241i)6-s + 0.381i·7-s + (0.219 + 0.975i)8-s + (0.345 + 0.597i)9-s + (−0.402 − 0.194i)10-s − 1.36i·11-s + (0.204 − 0.517i)12-s + (1.28 − 0.740i)13-s + (0.380 − 0.0280i)14-s + (0.124 + 0.215i)15-s + (0.956 − 0.290i)16-s + (0.526 − 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.417 + 0.908i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.417 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01560 - 0.650702i\)
\(L(\frac12)\) \(\approx\) \(1.01560 - 0.650702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.104 + 1.41i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-2.83 - 3.30i)T \)
good3 \( 1 + (0.481 - 0.834i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.00iT - 7T^{2} \)
11 \( 1 + 4.53iT - 11T^{2} \)
13 \( 1 + (-4.62 + 2.66i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.17 + 3.76i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.38 - 1.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.30 + 1.91i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.27T + 31T^{2} \)
37 \( 1 + 3.89iT - 37T^{2} \)
41 \( 1 + (0.221 + 0.127i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.13 + 0.653i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.20 - 3.00i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.70 - 5.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.64 + 8.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.67 - 8.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.09 - 3.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.75 + 3.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.39 - 12.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.03 - 8.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.98iT - 83T^{2} \)
89 \( 1 + (0.0672 - 0.0388i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.0 + 8.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.20162417458019828311635579899, −10.32629273495974467656443664595, −9.641430367134083447445574493809, −8.536196931387703325036514923358, −7.941936015460786862180589099008, −5.86714678136562925239471525568, −5.28861366398166043729819131011, −3.98511197824545105360823024349, −2.90651833324130410837137461124, −1.11829261391171044510357349123, 1.38331327985489741141524307365, 3.70759368717618567120615851180, 4.75268400202298636054358310266, 6.20919809261109852413882435100, 6.67653507969133424232588601396, 7.51816974599817881841136933436, 8.589262946750893683954075911988, 9.678610669171913452603774587439, 10.30417728687805361423321695430, 11.65778879739371935592775835156

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