↑

Properties

Label 2-380-19.11-c1-0-0
Degree $2$
Conductor $380$
Sign $-0.685 + 0.727i$
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  + (−1.58 + 2.74i)3-s + (−0.5 + 0.866i)5-s − 1.53·7-s + (−3.50 − 6.07i)9-s − 3.79·11-s + (3.34 + 5.80i)13-s + (−1.58 − 2.74i)15-s + (3.00 − 5.21i)17-s + (−3.93 − 1.87i)19-s + (2.42 − 4.20i)21-s + (−2.19 − 3.79i)23-s + (−0.499 − 0.866i)25-s + 12.7·27-s + (0.549 + 0.951i)29-s − 1.05·31-s + ⋯
L(s)  = 1  + (−0.913 + 1.58i)3-s + (−0.223 + 0.387i)5-s − 0.579·7-s + (−1.16 − 2.02i)9-s − 1.14·11-s + (0.928 + 1.60i)13-s + (−0.408 − 0.707i)15-s + (0.729 − 1.26i)17-s + (−0.902 − 0.430i)19-s + (0.529 − 0.917i)21-s + (−0.457 − 0.791i)23-s + (−0.0999 − 0.173i)25-s + 2.44·27-s + (0.102 + 0.176i)29-s − 0.189·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.685 + 0.727i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.685 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111114 - 0.257273i\)
\(L(\frac12)\) \(\approx\) \(0.111114 - 0.257273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (3.93 + 1.87i)T \)
good3 \( 1 + (1.58 - 2.74i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.53T + 7T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + (-3.34 - 5.80i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.00 + 5.21i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.19 + 3.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.549 - 0.951i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.05T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + (1.76 - 3.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.77 - 4.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.61 - 9.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.788 + 1.36i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.61 - 2.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0331 + 0.0574i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.85 - 4.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.23 + 5.60i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.98 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.996 + 1.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (1.53 + 2.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.43 - 7.68i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.56855376094811628755252929275, −10.95482668275405257092378838316, −10.20886407404502965335843857916, −9.457147976804104333665114783462, −8.554424856127191727218246199380, −6.92255625672662485783436228585, −6.07749006696308506506546165560, −4.97335775724522582939336293389, −4.12147879369052448686279277757, −2.99255292093521569468510521869, 0.20270259148807527191702159850, 1.75420256348134672667144594515, 3.42886313477209302230054869207, 5.47883685166616947994653171285, 5.80103369197580214797810669000, 6.94960828908668763769569110666, 8.037257838540144677329752144491, 8.342015707111867950259108745366, 10.35276284153740031403287449286, 10.72142848083957710274915151990

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