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Properties

Label 2-380-380.259-c1-0-4
Degree $2$
Conductor $380$
Sign $-0.504 - 0.863i$
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  + (−1.40 − 0.197i)2-s + (0.952 − 0.549i)3-s + (1.92 + 0.553i)4-s + (−2.22 − 0.224i)5-s + (−1.44 + 0.581i)6-s − 1.22·7-s + (−2.58 − 1.15i)8-s + (−0.895 + 1.55i)9-s + (3.07 + 0.753i)10-s + 0.0812i·11-s + (2.13 − 0.529i)12-s + (−1.99 + 3.45i)13-s + (1.71 + 0.242i)14-s + (−2.24 + 1.00i)15-s + (3.38 + 2.12i)16-s + (−4.46 + 2.57i)17-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.549 − 0.317i)3-s + (0.960 + 0.276i)4-s + (−0.994 − 0.100i)5-s + (−0.588 + 0.237i)6-s − 0.463·7-s + (−0.912 − 0.408i)8-s + (−0.298 + 0.516i)9-s + (0.971 + 0.238i)10-s + 0.0245i·11-s + (0.616 − 0.152i)12-s + (−0.553 + 0.958i)13-s + (0.459 + 0.0647i)14-s + (−0.579 + 0.260i)15-s + (0.846 + 0.531i)16-s + (−1.08 + 0.625i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.504 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162022 + 0.282358i\)
\(L(\frac12)\) \(\approx\) \(0.162022 + 0.282358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.197i)T \)
5 \( 1 + (2.22 + 0.224i)T \)
19 \( 1 + (2.32 - 3.68i)T \)
good3 \( 1 + (-0.952 + 0.549i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.22T + 7T^{2} \)
11 \( 1 - 0.0812iT - 11T^{2} \)
13 \( 1 + (1.99 - 3.45i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.46 - 2.57i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.94 + 6.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.920 + 0.531i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.146T + 31T^{2} \)
37 \( 1 + 2.88T + 37T^{2} \)
41 \( 1 + (10.5 - 6.07i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.50 - 6.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.81 - 4.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.68 + 6.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.76 + 3.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.88 + 3.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.93 - 5.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.376 - 0.652i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.44 + 0.833i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.08 + 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.81T + 83T^{2} \)
89 \( 1 + (0.325 + 0.187i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.65 + 4.59i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.43092047348013693130619993229, −10.79927896894384569528248800883, −9.706893669623153607417472440882, −8.574578757099005128999715117920, −8.298777703738836308234759406354, −7.13962372900003095929015149557, −6.46193428065191648696940589388, −4.55496147354464125038159824265, −3.19498669539376145283892335786, −1.98087239464125114928129566745, 0.25988030087346195040976085893, 2.68347793235063798563151731496, 3.59251535934827172122250902866, 5.23713161136201575709058584816, 6.74017937075195925412889115666, 7.36439745504851404108696380774, 8.510061522999415956280871532722, 9.037721098991056263888349485414, 9.963689573322617673819322003926, 10.96237989510839418440296593743

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