Properties

Label 2-380-380.159-c2-0-106
Degree $2$
Conductor $380$
Sign $0.177 - 0.984i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.82 − 0.808i)2-s + (−0.866 − 1.50i)3-s + (2.69 + 2.95i)4-s + (−4.28 − 2.56i)5-s + (0.371 + 3.44i)6-s − 10.8·7-s + (−2.53 − 7.58i)8-s + (2.99 − 5.19i)9-s + (5.76 + 8.16i)10-s − 20.5i·11-s + (2.10 − 6.60i)12-s + (4.62 + 2.67i)13-s + (19.7 + 8.75i)14-s + (−0.140 + 8.66i)15-s + (−1.50 + 15.9i)16-s + (−1.86 + 1.07i)17-s + ⋯
L(s)  = 1  + (−0.914 − 0.404i)2-s + (−0.288 − 0.500i)3-s + (0.672 + 0.739i)4-s + (−0.857 − 0.513i)5-s + (0.0618 + 0.574i)6-s − 1.54·7-s + (−0.316 − 0.948i)8-s + (0.333 − 0.577i)9-s + (0.576 + 0.816i)10-s − 1.87i·11-s + (0.175 − 0.550i)12-s + (0.356 + 0.205i)13-s + (1.41 + 0.625i)14-s + (−0.00934 + 0.577i)15-s + (−0.0942 + 0.995i)16-s + (−0.109 + 0.0634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.177 - 0.984i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.177 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0184837 + 0.0154448i\)
\(L(\frac12)\) \(\approx\) \(0.0184837 + 0.0154448i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.82 + 0.808i)T \)
5 \( 1 + (4.28 + 2.56i)T \)
19 \( 1 + (12.0 - 14.6i)T \)
good3 \( 1 + (0.866 + 1.50i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + 10.8T + 49T^{2} \)
11 \( 1 + 20.5iT - 121T^{2} \)
13 \( 1 + (-4.62 - 2.67i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (1.86 - 1.07i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (-12.9 + 22.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (20.7 - 35.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 1.42iT - 961T^{2} \)
37 \( 1 + 21.8iT - 1.36e3T^{2} \)
41 \( 1 + (-14.3 - 24.9i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (6.90 + 11.9i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (34.2 - 59.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-51.9 - 29.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (49.1 - 28.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (45.8 - 79.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-36.5 + 63.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-80.7 + 46.6i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-69.4 + 40.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-53.2 + 30.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 19.9T + 6.88e3T^{2} \)
89 \( 1 + (-56.0 + 96.9i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (2.63 - 1.51i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.58377536127038352107511477476, −9.240377456962838139111693421463, −8.796274628178890312121351681893, −7.73926787645993771124723541797, −6.63660478583256052125709093249, −6.05753190055606149617871392238, −3.83551201736262703092701310878, −3.15968382281934659059692502504, −1.01116479841122997240933563742, −0.01844361215092737780011662930, 2.34301370911763141533729337307, 3.87015550106652303155195514596, 5.13178869349818221966278134326, 6.58517797538841168564106450399, 7.10463885208961018890323658264, 8.013006960822579274877268507722, 9.445119287618906558455256969401, 9.858631129718874852031357443919, 10.69652321562074507144175596192, 11.51525934194099340275673197838

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