Properties

Label 2-380-380.239-c2-0-49
Degree $2$
Conductor $380$
Sign $0.997 + 0.0636i$
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  + (−1.73 − 0.996i)2-s + (1.05 − 1.82i)3-s + (2.01 + 3.45i)4-s + (2.78 + 4.15i)5-s + (−3.64 + 2.11i)6-s + 8.55·7-s + (−0.0476 − 7.99i)8-s + (2.28 + 3.95i)9-s + (−0.684 − 9.97i)10-s − 10.2i·11-s + (8.42 − 0.0334i)12-s + (−10.2 + 5.89i)13-s + (−14.8 − 8.52i)14-s + (10.5 − 0.700i)15-s + (−7.88 + 13.9i)16-s + (16.0 + 9.28i)17-s + ⋯
L(s)  = 1  + (−0.867 − 0.498i)2-s + (0.351 − 0.608i)3-s + (0.503 + 0.864i)4-s + (0.556 + 0.830i)5-s + (−0.607 + 0.352i)6-s + 1.22·7-s + (−0.00596 − 0.999i)8-s + (0.253 + 0.438i)9-s + (−0.0684 − 0.997i)10-s − 0.928i·11-s + (0.702 − 0.00279i)12-s + (−0.785 + 0.453i)13-s + (−1.05 − 0.609i)14-s + (0.700 − 0.0466i)15-s + (−0.493 + 0.869i)16-s + (0.946 + 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0636i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.997 + 0.0636i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.997 + 0.0636i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60149 - 0.0510453i\)
\(L(\frac12)\) \(\approx\) \(1.60149 - 0.0510453i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + 0.996i)T \)
5 \( 1 + (-2.78 - 4.15i)T \)
19 \( 1 + (14.5 - 12.2i)T \)
good3 \( 1 + (-1.05 + 1.82i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 - 8.55T + 49T^{2} \)
11 \( 1 + 10.2iT - 121T^{2} \)
13 \( 1 + (10.2 - 5.89i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-16.0 - 9.28i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (-18.2 - 31.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (6.36 + 11.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 8.11iT - 961T^{2} \)
37 \( 1 + 43.7iT - 1.36e3T^{2} \)
41 \( 1 + (19.1 - 33.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-29.0 + 50.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (21.4 + 37.0i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (1.54 - 0.890i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-68.8 - 39.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (41.5 + 71.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (20.7 + 35.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-79.6 - 45.9i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-85.6 - 49.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (15.0 + 8.68i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 144.T + 6.88e3T^{2} \)
89 \( 1 + (9.75 + 16.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-14.4 - 8.36i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.97981377197863380625023309394, −10.35587348389655643632222973511, −9.317595315226989664987310129845, −8.191757530991086458845416189366, −7.66221911000189307509764903760, −6.76521074880008363442002077148, −5.40700700318248551520839347929, −3.64474567932736878265937720539, −2.29448860917171750345259651860, −1.49573524573919936834142576034, 1.03505364197919189706724373844, 2.41857598101727875200554359791, 4.69999460875973274512064147656, 5.03127091896561863251918892446, 6.54289964023690395654180096974, 7.64896105068075924172783980760, 8.504806134918931059821059113986, 9.302323536073140515289989570673, 9.959849475294665957787159452474, 10.75210269924876767134798092770

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