Properties

Label 2-380-380.227-c2-0-94
Degree $2$
Conductor $380$
Sign $0.167 + 0.985i$
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  + (1.74 − 0.968i)2-s + (−1.12 + 1.12i)3-s + (2.12 − 3.38i)4-s + (3.50 − 3.57i)5-s + (−0.875 + 3.04i)6-s + (8.74 − 8.74i)7-s + (0.434 − 7.98i)8-s + 6.48i·9-s + (2.66 − 9.63i)10-s + 15.3i·11-s + (1.41 + 6.17i)12-s + (−16.4 − 16.4i)13-s + (6.82 − 23.7i)14-s + (0.0782 + 7.92i)15-s + (−6.97 − 14.3i)16-s + (−7.11 − 7.11i)17-s + ⋯
L(s)  = 1  + (0.874 − 0.484i)2-s + (−0.373 + 0.373i)3-s + (0.531 − 0.847i)4-s + (0.700 − 0.714i)5-s + (−0.145 + 0.507i)6-s + (1.24 − 1.24i)7-s + (0.0542 − 0.998i)8-s + 0.721i·9-s + (0.266 − 0.963i)10-s + 1.39i·11-s + (0.118 + 0.514i)12-s + (−1.26 − 1.26i)13-s + (0.487 − 1.69i)14-s + (0.00521 + 0.528i)15-s + (−0.436 − 0.899i)16-s + (−0.418 − 0.418i)17-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.30337 - 1.94451i\)
\(L(\frac12)\) \(\approx\) \(2.30337 - 1.94451i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.74 + 0.968i)T \)
5 \( 1 + (-3.50 + 3.57i)T \)
19 \( 1 + (-15.4 - 11.0i)T \)
good3 \( 1 + (1.12 - 1.12i)T - 9iT^{2} \)
7 \( 1 + (-8.74 + 8.74i)T - 49iT^{2} \)
11 \( 1 - 15.3iT - 121T^{2} \)
13 \( 1 + (16.4 + 16.4i)T + 169iT^{2} \)
17 \( 1 + (7.11 + 7.11i)T + 289iT^{2} \)
23 \( 1 + (-10.8 - 10.8i)T + 529iT^{2} \)
29 \( 1 - 37.4T + 841T^{2} \)
31 \( 1 + 27.7T + 961T^{2} \)
37 \( 1 + (-34.5 + 34.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 53.3iT - 1.68e3T^{2} \)
43 \( 1 + (4.74 + 4.74i)T + 1.84e3iT^{2} \)
47 \( 1 + (19.9 - 19.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (-25.1 - 25.1i)T + 2.80e3iT^{2} \)
59 \( 1 - 24.6iT - 3.48e3T^{2} \)
61 \( 1 - 23.0T + 3.72e3T^{2} \)
67 \( 1 + (-14.7 - 14.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 22.0T + 5.04e3T^{2} \)
73 \( 1 + (83.5 - 83.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 91.5iT - 6.24e3T^{2} \)
83 \( 1 + (-22.4 - 22.4i)T + 6.88e3iT^{2} \)
89 \( 1 - 101.T + 7.92e3T^{2} \)
97 \( 1 + (50.3 - 50.3i)T - 9.40e3iT^{2} \)
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

−10.92041459308186252021695769153, −10.13996016733976122606048536244, −9.709922547175169726006771381222, −7.84603694860370311575930031328, −7.19417934444175797542378861204, −5.47632419496814208949300197105, −4.89157088393767343336028478236, −4.36235694655454219883817454665, −2.39745712365589551030708186156, −1.16825806894658713392371297985, 1.99639775871443098233867062068, 3.05410925636457540406707132647, 4.75448245010594280884442547831, 5.63179054193934970076646116565, 6.40774324485216206588706261697, 7.21118879413847019099702877456, 8.509009741133067153095793198539, 9.249706354067150414057080706008, 10.90328894968500663139346381583, 11.67423603229164942556020784101

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