Properties

Label 2-380-380.159-c2-0-33
Degree $2$
Conductor $380$
Sign $0.0922 - 0.995i$
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  + (−1.94 + 0.482i)2-s + (0.480 + 0.831i)3-s + (3.53 − 1.87i)4-s + (4.92 + 0.849i)5-s + (−1.33 − 1.38i)6-s − 5.58·7-s + (−5.95 + 5.34i)8-s + (4.03 − 6.99i)9-s + (−9.97 + 0.730i)10-s + 16.9i·11-s + (3.25 + 2.03i)12-s + (4.31 + 2.48i)13-s + (10.8 − 2.69i)14-s + (1.65 + 4.50i)15-s + (8.97 − 13.2i)16-s + (−20.3 + 11.7i)17-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (0.160 + 0.277i)3-s + (0.883 − 0.468i)4-s + (0.985 + 0.169i)5-s + (−0.222 − 0.230i)6-s − 0.797·7-s + (−0.744 + 0.667i)8-s + (0.448 − 0.777i)9-s + (−0.997 + 0.0730i)10-s + 1.54i·11-s + (0.271 + 0.169i)12-s + (0.331 + 0.191i)13-s + (0.773 − 0.192i)14-s + (0.110 + 0.300i)15-s + (0.561 − 0.827i)16-s + (−1.19 + 0.692i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0922 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0922 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0922 - 0.995i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.0922 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.891514 + 0.812709i\)
\(L(\frac12)\) \(\approx\) \(0.891514 + 0.812709i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.94 - 0.482i)T \)
5 \( 1 + (-4.92 - 0.849i)T \)
19 \( 1 + (-14.2 - 12.5i)T \)
good3 \( 1 + (-0.480 - 0.831i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + 5.58T + 49T^{2} \)
11 \( 1 - 16.9iT - 121T^{2} \)
13 \( 1 + (-4.31 - 2.48i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (20.3 - 11.7i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (-1.57 + 2.73i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-15.8 + 27.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 - 7.75iT - 961T^{2} \)
37 \( 1 + 28.8iT - 1.36e3T^{2} \)
41 \( 1 + (-29.1 - 50.4i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-19.7 - 34.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (41.8 - 72.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-87.8 - 50.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (78.2 - 45.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (35.1 - 60.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-58.0 + 100. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-74.5 + 43.0i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (13.1 - 7.56i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (109. - 63.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 22.6T + 6.88e3T^{2} \)
89 \( 1 + (5.95 - 10.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-31.5 + 18.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.96758316170249959743120429717, −10.00626413373597608086982803409, −9.615567401618396512628713707344, −8.963718344094190084363755119065, −7.56950455274047928798200363192, −6.56605390670932472533020871180, −6.07824810075376508290616046347, −4.41458434996552408260227660113, −2.76822104115324215232753179167, −1.48541388117076455111418855515, 0.75147324161598297675252543431, 2.25983263661094996708034748487, 3.28586166372009789396979250324, 5.27534699119487353884923723851, 6.43046713114983628139011763525, 7.13322694231121489566064136406, 8.483212689412221543609957032654, 9.010635170075769177953545919925, 9.976820847830267497771548412127, 10.75344151426647419684567053404

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