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Properties

Label 2-380-76.27-c1-0-31
Degree $2$
Conductor $380$
Sign $0.938 + 0.345i$
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  + (1.19 + 0.758i)2-s + (1.50 − 2.60i)3-s + (0.849 + 1.81i)4-s + (−0.5 + 0.866i)5-s + (3.77 − 1.96i)6-s − 2.59i·7-s + (−0.359 + 2.80i)8-s + (−3.02 − 5.24i)9-s + (−1.25 + 0.654i)10-s + 4.32i·11-s + (5.99 + 0.510i)12-s + (5.09 − 2.94i)13-s + (1.97 − 3.10i)14-s + (1.50 + 2.60i)15-s + (−2.55 + 3.07i)16-s + (−0.759 + 1.31i)17-s + ⋯
L(s)  = 1  + (0.844 + 0.536i)2-s + (0.868 − 1.50i)3-s + (0.424 + 0.905i)4-s + (−0.223 + 0.387i)5-s + (1.54 − 0.804i)6-s − 0.982i·7-s + (−0.126 + 0.991i)8-s + (−1.00 − 1.74i)9-s + (−0.396 + 0.206i)10-s + 1.30i·11-s + (1.73 + 0.147i)12-s + (1.41 − 0.816i)13-s + (0.526 − 0.829i)14-s + (0.388 + 0.672i)15-s + (−0.639 + 0.769i)16-s + (−0.184 + 0.319i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.938 + 0.345i$
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: \((2,\ 380,\ (\ :1/2),\ 0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63189 - 0.469441i\)
\(L(\frac12)\) \(\approx\) \(2.63189 - 0.469441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 0.758i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-1.56 + 4.07i)T \)
good3 \( 1 + (-1.50 + 2.60i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 2.59iT - 7T^{2} \)
11 \( 1 - 4.32iT - 11T^{2} \)
13 \( 1 + (-5.09 + 2.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.759 - 1.31i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.53 - 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.10 - 4.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.63T + 31T^{2} \)
37 \( 1 - 5.86iT - 37T^{2} \)
41 \( 1 + (-0.487 - 0.281i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.31 - 2.49i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.11 + 0.643i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.04 + 1.75i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.06 - 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.65 - 2.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.662 - 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.864 - 1.49i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.793 + 1.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.810 - 1.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.0iT - 83T^{2} \)
89 \( 1 + (-13.2 + 7.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.46 - 5.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.62897865557434345539280438134, −10.65789968467760299412272650341, −9.067378299098796442484774105739, −7.994303070525303056641907845695, −7.37004648145322700329216748507, −6.87538363669674878203632024151, −5.77836278180331486918911777773, −4.06363640680447114982905774877, −3.14392805673761008051995219254, −1.72908932798587549922687601978, 2.22672409907321867801785766571, 3.63065911477845228129556344606, 3.97756012453570039408238399510, 5.40664927212797499836588742270, 6.01016667437506391382833582665, 8.085360209196831062638686753404, 9.026530172310750960138796234344, 9.427477806737391682589898956349, 10.73484223538700477795324338835, 11.23465321250174185950778216956

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