Properties

Label 2-380-380.227-c2-0-23
Degree $2$
Conductor $380$
Sign $-0.985 + 0.167i$
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  + (0.657 + 1.88i)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 + (−6.75 − 4.29i)8-s + 8.53i·9-s + (−8.40 + 5.41i)10-s − 5.36i·11-s + (−0.315 + 2.71i)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  + (0.328 + 0.944i)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 + (−0.843 − 0.536i)8-s + 0.948i·9-s + (−0.840 + 0.541i)10-s − 0.487i·11-s + (−0.0262 + 0.226i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.985 + 0.167i$
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: \((2,\ 380,\ (\ :1),\ -0.985 + 0.167i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.118443 - 1.40168i\)
\(L(\frac12)\) \(\approx\) \(0.118443 - 1.40168i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.657 - 1.88i)T \)
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

−11.54235966183790910860257246536, −10.79700167545910438733905815513, −9.581333572829135454245400310540, −8.650380803476619493978425087606, −7.76329544559228604175172056059, −6.72463426856789623670353381992, −6.15305724181390737301448841731, −4.90855114285061867821309035739, −3.64350503321676803172436862009, −2.36705130635630813606218066541, 0.54028122856549065249694843958, 1.98550363487051145801584980289, 3.60530622695261946379471607248, 4.36466772145863479434606165762, 5.57485002489863535899347166232, 6.55012070442504201260146455314, 8.381615415308820783927676958016, 8.925236619115892849004422764746, 9.855825823750403939560256007957, 10.61080063893038128878689598385

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