Properties

Label 2-380-380.227-c2-0-80
Degree $2$
Conductor $380$
Sign $0.461 + 0.886i$
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  + (−1.99 + 0.165i)2-s + (2.36 − 2.36i)3-s + (3.94 − 0.661i)4-s + (2.79 + 4.14i)5-s + (−4.31 + 5.10i)6-s + (8.32 − 8.32i)7-s + (−7.75 + 1.97i)8-s − 2.16i·9-s + (−6.26 − 7.79i)10-s − 14.7i·11-s + (7.75 − 10.8i)12-s + (−4.62 − 4.62i)13-s + (−15.2 + 17.9i)14-s + (16.4 + 3.17i)15-s + (15.1 − 5.21i)16-s + (2.68 + 2.68i)17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0829i)2-s + (0.787 − 0.787i)3-s + (0.986 − 0.165i)4-s + (0.559 + 0.828i)5-s + (−0.719 + 0.850i)6-s + (1.18 − 1.18i)7-s + (−0.969 + 0.246i)8-s − 0.240i·9-s + (−0.626 − 0.779i)10-s − 1.33i·11-s + (0.646 − 0.906i)12-s + (−0.355 − 0.355i)13-s + (−1.08 + 1.28i)14-s + (1.09 + 0.211i)15-s + (0.945 − 0.325i)16-s + (0.157 + 0.157i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.461 + 0.886i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.461 + 0.886i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.48036 - 0.898208i\)
\(L(\frac12)\) \(\approx\) \(1.48036 - 0.898208i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.165i)T \)
5 \( 1 + (-2.79 - 4.14i)T \)
19 \( 1 + (12.7 + 14.0i)T \)
good3 \( 1 + (-2.36 + 2.36i)T - 9iT^{2} \)
7 \( 1 + (-8.32 + 8.32i)T - 49iT^{2} \)
11 \( 1 + 14.7iT - 121T^{2} \)
13 \( 1 + (4.62 + 4.62i)T + 169iT^{2} \)
17 \( 1 + (-2.68 - 2.68i)T + 289iT^{2} \)
23 \( 1 + (-19.0 - 19.0i)T + 529iT^{2} \)
29 \( 1 - 25.8T + 841T^{2} \)
31 \( 1 + 39.2T + 961T^{2} \)
37 \( 1 + (-20.2 + 20.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 2.81iT - 1.68e3T^{2} \)
43 \( 1 + (32.2 + 32.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (20.7 - 20.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-6.53 - 6.53i)T + 2.80e3iT^{2} \)
59 \( 1 + 14.4iT - 3.48e3T^{2} \)
61 \( 1 - 73.3T + 3.72e3T^{2} \)
67 \( 1 + (-71.1 - 71.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 90.3T + 5.04e3T^{2} \)
73 \( 1 + (30.6 - 30.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 123. iT - 6.24e3T^{2} \)
83 \( 1 + (75.5 + 75.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 56.2T + 7.92e3T^{2} \)
97 \( 1 + (99.3 - 99.3i)T - 9.40e3iT^{2} \)
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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

−10.99021822040075956710938207927, −10.14143830191051358303569940138, −8.883799264425994563469850536829, −8.121728688318878877528814430746, −7.38238555175499667221527232724, −6.74999210613990623748191613816, −5.39520124424722761245816007157, −3.34836618452010627326640805651, −2.21322038328211546843354781634, −1.04658562592499347375676726709, 1.69526477806783589823000923184, 2.55171418925624264774908724992, 4.39472347183490345729221317387, 5.33207757494358192762759110041, 6.73811168893363049143641775280, 8.243535853798424564244479234093, 8.526307059646575598017397948047, 9.522499360414365584716501399778, 9.871232850832506273256988721017, 11.10149085714340343600407861317

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