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Properties

Label 2-380-380.179-c1-0-48
Degree $2$
Conductor $380$
Sign $-0.928 + 0.370i$
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  + (−0.204 − 1.39i)2-s + (1.49 + 0.861i)3-s + (−1.91 + 0.572i)4-s + (−1.59 − 1.56i)5-s + (0.900 − 2.26i)6-s − 2.20·7-s + (1.19 + 2.56i)8-s + (−0.0139 − 0.0241i)9-s + (−1.87 + 2.54i)10-s − 4.26i·11-s + (−3.35 − 0.797i)12-s + (−2.52 − 4.37i)13-s + (0.449 + 3.07i)14-s + (−1.02 − 3.71i)15-s + (3.34 − 2.19i)16-s + (−2.47 − 1.42i)17-s + ⋯
L(s)  = 1  + (−0.144 − 0.989i)2-s + (0.861 + 0.497i)3-s + (−0.958 + 0.286i)4-s + (−0.712 − 0.702i)5-s + (0.367 − 0.924i)6-s − 0.831·7-s + (0.421 + 0.906i)8-s + (−0.00465 − 0.00805i)9-s + (−0.591 + 0.806i)10-s − 1.28i·11-s + (−0.968 − 0.230i)12-s + (−0.700 − 1.21i)13-s + (0.120 + 0.823i)14-s + (−0.264 − 0.959i)15-s + (0.836 − 0.548i)16-s + (−0.599 − 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.928 + 0.370i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.928 + 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164623 - 0.856351i\)
\(L(\frac12)\) \(\approx\) \(0.164623 - 0.856351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.204 + 1.39i)T \)
5 \( 1 + (1.59 + 1.56i)T \)
19 \( 1 + (-3.03 - 3.12i)T \)
good3 \( 1 + (-1.49 - 0.861i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + 2.20T + 7T^{2} \)
11 \( 1 + 4.26iT - 11T^{2} \)
13 \( 1 + (2.52 + 4.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.47 + 1.42i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.151 + 0.261i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.15 + 2.97i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.96T + 31T^{2} \)
37 \( 1 - 5.24T + 37T^{2} \)
41 \( 1 + (-8.43 - 4.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.03 - 8.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.14 + 5.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.08 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.530 + 0.918i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.60 + 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.69 + 2.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.49 - 4.32i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.80 - 2.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.31 + 5.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.63T + 83T^{2} \)
89 \( 1 + (-10.0 + 5.81i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.77 + 8.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.95755956987474273266018767792, −9.822600317815744547189391516888, −9.310567732904981959344124860634, −8.365082820313100570646569435279, −7.80675093556099497997091407111, −5.85712998921484523000693329631, −4.58057606277672630309440580024, −3.43809064035986323525002517556, −2.92161402523525016144551198468, −0.54797541537152655681428136789, 2.35104072211727146387122951659, 3.78026019689550236795803907692, 4.89346617859867732265173481518, 6.61007509628311422986542719704, 7.10648503787169717787705806043, 7.73804321510721400315473534133, 8.927148328172799827838141069286, 9.538077120823497426043064045046, 10.61800801482535717180515196300, 11.96360272029156203715625408772

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