↑

Properties

Label 2-380-76.27-c1-0-16
Degree $2$
Conductor $380$
Sign $0.962 + 0.272i$
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  + (−0.916 − 1.07i)2-s + (−1.57 + 2.72i)3-s + (−0.319 + 1.97i)4-s + (0.5 − 0.866i)5-s + (4.37 − 0.803i)6-s − 3.91i·7-s + (2.41 − 1.46i)8-s + (−3.45 − 5.97i)9-s + (−1.39 + 0.255i)10-s + 3.87i·11-s + (−4.87 − 3.97i)12-s + (2.18 − 1.26i)13-s + (−4.21 + 3.58i)14-s + (1.57 + 2.72i)15-s + (−3.79 − 1.26i)16-s + (1.49 − 2.58i)17-s + ⋯
L(s)  = 1  + (−0.648 − 0.761i)2-s + (−0.908 + 1.57i)3-s + (−0.159 + 0.987i)4-s + (0.223 − 0.387i)5-s + (1.78 − 0.328i)6-s − 1.48i·7-s + (0.855 − 0.518i)8-s + (−1.15 − 1.99i)9-s + (−0.439 + 0.0807i)10-s + 1.16i·11-s + (−1.40 − 1.14i)12-s + (0.606 − 0.350i)13-s + (−1.12 + 0.959i)14-s + (0.406 + 0.703i)15-s + (−0.948 − 0.315i)16-s + (0.361 − 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.962 + 0.272i$
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: \((2,\ 380,\ (\ :1/2),\ 0.962 + 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724794 - 0.100712i\)
\(L(\frac12)\) \(\approx\) \(0.724794 - 0.100712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.916 + 1.07i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-1.32 - 4.15i)T \)
good3 \( 1 + (1.57 - 2.72i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 3.91iT - 7T^{2} \)
11 \( 1 - 3.87iT - 11T^{2} \)
13 \( 1 + (-2.18 + 1.26i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.49 + 2.58i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.25 + 3.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.45 + 2.57i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.02T + 31T^{2} \)
37 \( 1 - 8.10iT - 37T^{2} \)
41 \( 1 + (-8.63 - 4.98i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.84 + 2.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.04 + 1.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.11 + 2.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.86 + 3.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.01 + 5.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.14 + 5.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.55 + 6.15i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.54 - 2.67i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.20 + 2.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.29iT - 83T^{2} \)
89 \( 1 + (-11.4 + 6.62i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.73 + 2.15i)T + (48.5 + 84.0i)T^{2} \)
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

−11.00500506127325618181006961210, −10.22077154218528435928107587298, −9.968167848939503249065341695731, −9.030068382673154827387940464861, −7.77030679823545067362234805228, −6.56326400702062225282449742220, −4.94879608190689917250685016108, −4.33326262544295280620606595178, −3.33229101749358853502559800238, −0.893085269240424404007874833447, 1.13399530288725055389096613400, 2.55006511871039185716543977730, 5.42225313477001373237708592087, 5.80272542226035215380759676393, 6.61871633621730392992684716314, 7.46371551819788184780316955664, 8.549629371145823304476985094813, 9.113380583909311625064084862521, 10.84791282156530538612906776917, 11.24311576684780267254565253291

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