Properties

Label 2-380-20.19-c2-0-75
Degree $2$
Conductor $380$
Sign $-0.494 + 0.869i$
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 − 0.485i)2-s − 4.64·3-s + (3.52 − 1.88i)4-s + (2.66 + 4.22i)5-s + (−9.01 + 2.25i)6-s − 9.90·7-s + (5.92 − 5.37i)8-s + 12.6·9-s + (7.23 + 6.90i)10-s − 2.36i·11-s + (−16.4 + 8.76i)12-s − 16.5i·13-s + (−19.2 + 4.81i)14-s + (−12.4 − 19.6i)15-s + (8.89 − 13.3i)16-s − 21.4i·17-s + ⋯
L(s)  = 1  + (0.970 − 0.242i)2-s − 1.54·3-s + (0.881 − 0.471i)4-s + (0.533 + 0.845i)5-s + (−1.50 + 0.376i)6-s − 1.41·7-s + (0.741 − 0.671i)8-s + 1.40·9-s + (0.723 + 0.690i)10-s − 0.214i·11-s + (−1.36 + 0.730i)12-s − 1.27i·13-s + (−1.37 + 0.343i)14-s + (−0.826 − 1.31i)15-s + (0.555 − 0.831i)16-s − 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.494 + 0.869i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.494 + 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.567448 - 0.975557i\)
\(L(\frac12)\) \(\approx\) \(0.567448 - 0.975557i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.94 + 0.485i)T \)
5 \( 1 + (-2.66 - 4.22i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 + 4.64T + 9T^{2} \)
7 \( 1 + 9.90T + 49T^{2} \)
11 \( 1 + 2.36iT - 121T^{2} \)
13 \( 1 + 16.5iT - 169T^{2} \)
17 \( 1 + 21.4iT - 289T^{2} \)
23 \( 1 + 6.86T + 529T^{2} \)
29 \( 1 + 36.1T + 841T^{2} \)
31 \( 1 + 46.4iT - 961T^{2} \)
37 \( 1 + 17.7iT - 1.36e3T^{2} \)
41 \( 1 + 54.6T + 1.68e3T^{2} \)
43 \( 1 + 4.27T + 1.84e3T^{2} \)
47 \( 1 - 40.9T + 2.20e3T^{2} \)
53 \( 1 - 10.7iT - 2.80e3T^{2} \)
59 \( 1 + 70.4iT - 3.48e3T^{2} \)
61 \( 1 - 101.T + 3.72e3T^{2} \)
67 \( 1 + 7.29T + 4.48e3T^{2} \)
71 \( 1 - 31.8iT - 5.04e3T^{2} \)
73 \( 1 - 83.7iT - 5.32e3T^{2} \)
79 \( 1 - 137. iT - 6.24e3T^{2} \)
83 \( 1 + 99.8T + 6.88e3T^{2} \)
89 \( 1 - 76.9T + 7.92e3T^{2} \)
97 \( 1 + 155. iT - 9.40e3T^{2} \)
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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

−11.10207530907743115828543116988, −10.13479598619799041226350285885, −9.712693138022216271153561591618, −7.33036584041256795870312700233, −6.64385531908175067956316318919, −5.80795993599081123833819289452, −5.34067572651265271157878278704, −3.70634633168503008949128861815, −2.61872412681739047260439117902, −0.42254742652175364188387882348, 1.71099816402262103526597487698, 3.74161848929931119389929279375, 4.76602547520388145386992454327, 5.75083123216655556551669375018, 6.33308902070013393687691667257, 7.04457177820696002469793391143, 8.720234626899797021074298004216, 9.923851975178804831836702481004, 10.67210764390334600501281853782, 11.86227467822747942128801624513

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