Properties

Label 2-380-380.159-c2-0-111
Degree $2$
Conductor $380$
Sign $-0.659 - 0.751i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − 0.996i)2-s + (−1.05 − 1.82i)3-s + (2.01 − 3.45i)4-s + (−4.98 + 0.332i)5-s + (−3.64 − 2.11i)6-s − 8.55·7-s + (0.0476 − 7.99i)8-s + (2.28 − 3.95i)9-s + (−8.31 + 5.54i)10-s + 10.2i·11-s + (−8.42 − 0.0334i)12-s + (10.2 + 5.89i)13-s + (−14.8 + 8.52i)14-s + (5.86 + 8.75i)15-s + (−7.88 − 13.9i)16-s + (−16.0 + 9.28i)17-s + ⋯
L(s)  = 1  + (0.867 − 0.498i)2-s + (−0.351 − 0.608i)3-s + (0.503 − 0.864i)4-s + (−0.997 + 0.0664i)5-s + (−0.607 − 0.352i)6-s − 1.22·7-s + (0.00596 − 0.999i)8-s + (0.253 − 0.438i)9-s + (−0.831 + 0.554i)10-s + 0.928i·11-s + (−0.702 − 0.00279i)12-s + (0.785 + 0.453i)13-s + (−1.05 + 0.609i)14-s + (0.390 + 0.583i)15-s + (−0.493 − 0.869i)16-s + (−0.946 + 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.659 - 0.751i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.207803 + 0.458895i\)
\(L(\frac12)\) \(\approx\) \(0.207803 + 0.458895i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + 0.996i)T \)
5 \( 1 + (4.98 - 0.332i)T \)
19 \( 1 + (14.5 + 12.2i)T \)
good3 \( 1 + (1.05 + 1.82i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + 8.55T + 49T^{2} \)
11 \( 1 - 10.2iT - 121T^{2} \)
13 \( 1 + (-10.2 - 5.89i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (16.0 - 9.28i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (18.2 - 31.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (6.36 - 11.0i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 8.11iT - 961T^{2} \)
37 \( 1 + 43.7iT - 1.36e3T^{2} \)
41 \( 1 + (19.1 + 33.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (29.0 + 50.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-21.4 + 37.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-1.54 - 0.890i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-68.8 + 39.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (41.5 - 71.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-20.7 + 35.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-79.6 + 45.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (85.6 - 49.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (15.0 - 8.68i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 144.T + 6.88e3T^{2} \)
89 \( 1 + (9.75 - 16.9i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (14.4 - 8.36i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−10.88592117369470526040997353971, −9.850269313389760497120173647716, −8.854961185869551964100188016657, −7.11883859007113900455916546754, −6.80614505503758936239479526071, −5.76595922955054357322142279377, −4.16853374614705306637175717964, −3.61695529030726936022565393668, −1.94410189440293547781746118105, −0.16573946326527656702203344786, 2.93258428062692081995825383982, 3.90890538292661835428566479609, 4.69562056815625882041724113045, 6.03095117437490881051450055919, 6.66745395175189549074973384405, 7.994591666914614880703664953991, 8.647331760395693253804752743659, 10.14191631341949046402489832083, 11.01161706431252007154275878071, 11.68647634328561160283321308966

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