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Properties

Label 2-380-20.7-c1-0-31
Degree $2$
Conductor $380$
Sign $0.789 + 0.614i$
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  + (−1.36 − 0.378i)2-s + (1.58 + 1.58i)3-s + (1.71 + 1.03i)4-s + (−1.74 − 1.39i)5-s + (−1.55 − 2.75i)6-s + (1.94 − 1.94i)7-s + (−1.94 − 2.05i)8-s + 1.99i·9-s + (1.85 + 2.56i)10-s − 4.18i·11-s + (1.07 + 4.34i)12-s + (3.14 − 3.14i)13-s + (−3.38 + 1.91i)14-s + (−0.560 − 4.96i)15-s + (1.87 + 3.53i)16-s + (−2.53 − 2.53i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.267i)2-s + (0.912 + 0.912i)3-s + (0.856 + 0.515i)4-s + (−0.781 − 0.623i)5-s + (−0.635 − 1.12i)6-s + (0.734 − 0.734i)7-s + (−0.687 − 0.726i)8-s + 0.666i·9-s + (0.586 + 0.809i)10-s − 1.26i·11-s + (0.311 + 1.25i)12-s + (0.872 − 0.872i)13-s + (−0.904 + 0.511i)14-s + (−0.144 − 1.28i)15-s + (0.467 + 0.883i)16-s + (−0.616 − 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.789 + 0.614i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.789 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06175 - 0.364560i\)
\(L(\frac12)\) \(\approx\) \(1.06175 - 0.364560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.378i)T \)
5 \( 1 + (1.74 + 1.39i)T \)
19 \( 1 + T \)
good3 \( 1 + (-1.58 - 1.58i)T + 3iT^{2} \)
7 \( 1 + (-1.94 + 1.94i)T - 7iT^{2} \)
11 \( 1 + 4.18iT - 11T^{2} \)
13 \( 1 + (-3.14 + 3.14i)T - 13iT^{2} \)
17 \( 1 + (2.53 + 2.53i)T + 17iT^{2} \)
23 \( 1 + (-3.70 - 3.70i)T + 23iT^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 + 0.276iT - 31T^{2} \)
37 \( 1 + (-5.28 - 5.28i)T + 37iT^{2} \)
41 \( 1 + 5.13T + 41T^{2} \)
43 \( 1 + (7.75 + 7.75i)T + 43iT^{2} \)
47 \( 1 + (-3.74 + 3.74i)T - 47iT^{2} \)
53 \( 1 + (-6.04 + 6.04i)T - 53iT^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 + 6.42T + 61T^{2} \)
67 \( 1 + (-5.17 + 5.17i)T - 67iT^{2} \)
71 \( 1 - 3.20iT - 71T^{2} \)
73 \( 1 + (6.83 - 6.83i)T - 73iT^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 + (-4.21 - 4.21i)T + 83iT^{2} \)
89 \( 1 - 12.4iT - 89T^{2} \)
97 \( 1 + (8.49 + 8.49i)T + 97iT^{2} \)
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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

−11.01297373136098121020506381226, −10.39682118165081561823717797101, −9.152958117406943107425172588866, −8.550406354694764673756548105483, −8.091048731925415315008863584678, −6.94680152088696112943357155342, −5.16568911973112001341396314472, −3.79373471525460078179958615766, −3.17782734078507207065159104275, −1.01602169887233374517514682559, 1.78326815803080775800167171966, 2.59546847798801003812429426587, 4.41146989456293706144355102153, 6.27297229046319393722158638288, 7.02664281177612765013245129436, 7.88848158565433729297062650972, 8.457329007371674596940506980764, 9.246530184982631086717259659540, 10.54010368203475425414539306540, 11.42609363769088321592130007337

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