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Properties

Label 2-380-76.31-c1-0-24
Degree $2$
Conductor $380$
Sign $-0.858 + 0.512i$
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  + (−1.41 − 0.0746i)2-s + (−1.09 − 1.89i)3-s + (1.98 + 0.210i)4-s + (0.5 + 0.866i)5-s + (1.40 + 2.76i)6-s − 2.65i·7-s + (−2.79 − 0.445i)8-s + (−0.901 + 1.56i)9-s + (−0.641 − 1.26i)10-s − 1.44i·11-s + (−1.77 − 4.00i)12-s + (1.00 + 0.577i)13-s + (−0.197 + 3.74i)14-s + (1.09 − 1.89i)15-s + (3.91 + 0.838i)16-s + (−1.39 − 2.42i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0527i)2-s + (−0.632 − 1.09i)3-s + (0.994 + 0.105i)4-s + (0.223 + 0.387i)5-s + (0.573 + 1.12i)6-s − 1.00i·7-s + (−0.987 − 0.157i)8-s + (−0.300 + 0.520i)9-s + (−0.202 − 0.398i)10-s − 0.435i·11-s + (−0.513 − 1.15i)12-s + (0.277 + 0.160i)13-s + (−0.0529 + 1.00i)14-s + (0.282 − 0.490i)15-s + (0.977 + 0.209i)16-s + (−0.339 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.858 + 0.512i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.858 + 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150987 - 0.547815i\)
\(L(\frac12)\) \(\approx\) \(0.150987 - 0.547815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0746i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-4.25 + 0.941i)T \)
good3 \( 1 + (1.09 + 1.89i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.65iT - 7T^{2} \)
11 \( 1 + 1.44iT - 11T^{2} \)
13 \( 1 + (-1.00 - 0.577i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.39 + 2.42i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.05 + 1.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.87 + 2.81i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.90T + 31T^{2} \)
37 \( 1 + 7.08iT - 37T^{2} \)
41 \( 1 + (8.49 - 4.90i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.24 - 3.02i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.81 - 1.04i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.42 - 3.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.64 + 2.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.74 + 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.75 + 9.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.58 - 2.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.83 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.51 + 9.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.78iT - 83T^{2} \)
89 \( 1 + (-1.26 - 0.727i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.0 + 5.80i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.18979521900285450955545798778, −10.09964379068502425309478608506, −9.212330595143009576012319308511, −7.905206123399058610863801359490, −7.20176319523855976313928986514, −6.59190158758828278669880643465, −5.58242677476065338180028425145, −3.53955245491843538166407644103, −1.90100604943275927649444118818, −0.55599707756504765660385772100, 1.90962135682329666999162988509, 3.63548860978332085973050366639, 5.26412436955381716918773478555, 5.74708847857833586429766934778, 7.09633466981452705835958459169, 8.381037967217884907299944603038, 9.133960162883009867124036165987, 9.901262193059960521526587715652, 10.56726853755011417838225328876, 11.56662728535264734706622797450

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