Properties

Label 2-380-380.303-c2-0-101
Degree $2$
Conductor $380$
Sign $0.157 + 0.987i$
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  + (1.88 − 0.657i)2-s + (0.483 + 0.483i)3-s + (3.13 − 2.48i)4-s + (1.17 − 4.86i)5-s + (1.23 + 0.595i)6-s + (1.64 + 1.64i)7-s + (4.29 − 6.75i)8-s − 8.53i·9-s + (−0.977 − 9.95i)10-s − 5.36i·11-s + (2.71 + 0.315i)12-s + (−9.72 + 9.72i)13-s + (4.18 + 2.02i)14-s + (2.91 − 1.78i)15-s + (3.66 − 15.5i)16-s + (−13.8 + 13.8i)17-s + ⋯
L(s)  = 1  + (0.944 − 0.328i)2-s + (0.161 + 0.161i)3-s + (0.783 − 0.620i)4-s + (0.234 − 0.972i)5-s + (0.205 + 0.0992i)6-s + (0.235 + 0.235i)7-s + (0.536 − 0.843i)8-s − 0.948i·9-s + (−0.0977 − 0.995i)10-s − 0.487i·11-s + (0.226 + 0.0262i)12-s + (−0.748 + 0.748i)13-s + (0.299 + 0.144i)14-s + (0.194 − 0.118i)15-s + (0.229 − 0.973i)16-s + (−0.815 + 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.157 + 0.987i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.157 + 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.48488 - 2.11963i\)
\(L(\frac12)\) \(\approx\) \(2.48488 - 2.11963i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.88 + 0.657i)T \)
5 \( 1 + (-1.17 + 4.86i)T \)
19 \( 1 + (-18.2 + 5.23i)T \)
good3 \( 1 + (-0.483 - 0.483i)T + 9iT^{2} \)
7 \( 1 + (-1.64 - 1.64i)T + 49iT^{2} \)
11 \( 1 + 5.36iT - 121T^{2} \)
13 \( 1 + (9.72 - 9.72i)T - 169iT^{2} \)
17 \( 1 + (13.8 - 13.8i)T - 289iT^{2} \)
23 \( 1 + (0.900 - 0.900i)T - 529iT^{2} \)
29 \( 1 - 28.2T + 841T^{2} \)
31 \( 1 - 24.8T + 961T^{2} \)
37 \( 1 + (-13.7 - 13.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 33.6iT - 1.68e3T^{2} \)
43 \( 1 + (-18.0 + 18.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (-14.8 - 14.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (43.7 - 43.7i)T - 2.80e3iT^{2} \)
59 \( 1 - 91.5iT - 3.48e3T^{2} \)
61 \( 1 - 3.74T + 3.72e3T^{2} \)
67 \( 1 + (-11.4 + 11.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 129.T + 5.04e3T^{2} \)
73 \( 1 + (-17.7 - 17.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 36.2iT - 6.24e3T^{2} \)
83 \( 1 + (104. - 104. i)T - 6.88e3iT^{2} \)
89 \( 1 + 63.3T + 7.92e3T^{2} \)
97 \( 1 + (-52.4 - 52.4i)T + 9.40e3iT^{2} \)
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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

−11.25856175719263227152628466911, −9.973690205142769473551141417857, −9.267800940381076087304016052506, −8.269686699162805555212802102701, −6.81207623293616629692110607714, −5.92153149129776240121685426946, −4.83797621204042767161285193791, −4.03669383513008471275354633310, −2.62510301933021376065117704192, −1.13572580438687916858660590872, 2.20811071561406028833094779613, 3.04704043857723732662913318567, 4.56037203307260359823573696995, 5.41293915666111335453690378980, 6.66966979859685882913641716573, 7.42582088498522251925164463862, 8.068972417418045318463015519392, 9.737649357231564788310004528040, 10.65302874195412049693846672257, 11.38054826065872526074172894975

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