↑

Properties

Label 2-380-76.27-c1-0-21
Degree $2$
Conductor $380$
Sign $0.983 - 0.181i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.776 + 1.18i)2-s + (−1.45 + 2.51i)3-s + (−0.794 − 1.83i)4-s + (0.5 − 0.866i)5-s + (−1.84 − 3.66i)6-s − 1.19i·7-s + (2.78 + 0.485i)8-s + (−2.70 − 4.69i)9-s + (0.635 + 1.26i)10-s − 4.65i·11-s + (5.76 + 0.665i)12-s + (2.63 − 1.52i)13-s + (1.41 + 0.931i)14-s + (1.45 + 2.51i)15-s + (−2.73 + 2.91i)16-s + (1.05 − 1.82i)17-s + ⋯
L(s)  = 1  + (−0.548 + 0.835i)2-s + (−0.837 + 1.45i)3-s + (−0.397 − 0.917i)4-s + (0.223 − 0.387i)5-s + (−0.752 − 1.49i)6-s − 0.453i·7-s + (0.985 + 0.171i)8-s + (−0.903 − 1.56i)9-s + (0.200 + 0.399i)10-s − 1.40i·11-s + (1.66 + 0.192i)12-s + (0.732 − 0.422i)13-s + (0.378 + 0.248i)14-s + (0.374 + 0.648i)15-s + (−0.684 + 0.729i)16-s + (0.255 − 0.442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.983 - 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626747 + 0.0575013i\)
\(L(\frac12)\) \(\approx\) \(0.626747 + 0.0575013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.776 - 1.18i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.865 + 4.27i)T \)
good3 \( 1 + (1.45 - 2.51i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.19iT - 7T^{2} \)
11 \( 1 + 4.65iT - 11T^{2} \)
13 \( 1 + (-2.63 + 1.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.05 + 1.82i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (5.61 - 3.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.84 - 1.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.17T + 31T^{2} \)
37 \( 1 + 2.05iT - 37T^{2} \)
41 \( 1 + (1.59 + 0.920i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.54 - 0.891i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.0 + 6.36i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.74 - 3.89i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.35 - 9.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.43 + 5.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.67 + 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.55 + 7.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.11 + 8.85i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.55 + 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.9iT - 83T^{2} \)
89 \( 1 + (3.62 - 2.09i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.2 - 5.92i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.82176505101020931221789860169, −10.58108287372391875128284384933, −9.435550151669956252435941596235, −8.878485620995512422221565541473, −7.76941106744773562515644457802, −6.24298896227334683078375743106, −5.67329487922144015461895803820, −4.74531220673270273006083926235, −3.65207040476721595890512570359, −0.59797536696176159886658246233, 1.51373369282487894610363909003, 2.35959368743976850657330121211, 4.20776089106884448610246217692, 5.79764954602077614485350621819, 6.69043777392742318332187435129, 7.62999251478175156087134764312, 8.413470446051820922101667134175, 9.737988634170355869207605592798, 10.55555989395447638634837627764, 11.49222521728750409816505974074

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