↑

Properties

Label 2-3800-152.11-c0-0-0
Degree $2$
Conductor $3800$
Sign $-0.671 - 0.740i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 − 0.866i)4-s + (−0.999 − 1.73i)6-s + 0.999·8-s + (−1.49 − 2.59i)9-s − 11-s + 1.99·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + 3·18-s + 19-s + (0.5 − 0.866i)22-s + (−1 + 1.73i)24-s + 4·27-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 − 0.866i)4-s + (−0.999 − 1.73i)6-s + 0.999·8-s + (−1.49 − 2.59i)9-s − 11-s + 1.99·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + 3·18-s + 19-s + (0.5 − 0.866i)22-s + (−1 + 1.73i)24-s + 4·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.671 - 0.740i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5470633676\)
\(L(\frac12)\) \(\approx\) \(0.5470633676\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - 2T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.122071345926588598575568097106, −8.332069598149599117981018553842, −7.45468440700127506174417905295, −6.60107986511674122415307284124, −5.72537649953636185351604359329, −5.27696142126478775552119289126, −4.77086111356691073240131427652, −3.86393011855442051595249922593, −2.87508101750869351776484650408, −0.76022739601535963651202880162, 0.68423752526969071470027603493, 1.68287882945680099695377238636, 2.43647482493826202488866773414, 3.39005638050539678256863861898, 4.83243154849529916787784962448, 5.45949140196912992993251475685, 6.24465723971237284711489609820, 7.21711840564239426605320191311, 7.66431842077474198234987457278, 8.204904509616653978322091610269

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