↑

Properties

Label 2-40e2-16.5-c1-0-14
Degree $2$
Conductor $1600$
Sign $-0.110 - 0.993i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + (−1.66 + 1.66i)3-s + 1.87i·7-s − 2.56i·9-s + (3.29 + 3.29i)11-s + (1.90 − 1.90i)13-s + 2.57·17-s + (5.76 − 5.76i)19-s + (−3.12 − 3.12i)21-s + 7.58i·23-s + (−0.728 − 0.728i)27-s + (6.45 − 6.45i)29-s + 0.799·31-s − 10.9·33-s + (2.69 + 2.69i)37-s + 6.33i·39-s + ⋯
L(s)  = 1  + (−0.962 + 0.962i)3-s + 0.708i·7-s − 0.854i·9-s + (0.994 + 0.994i)11-s + (0.527 − 0.527i)13-s + 0.623·17-s + (1.32 − 1.32i)19-s + (−0.681 − 0.681i)21-s + 1.58i·23-s + (−0.140 − 0.140i)27-s + (1.19 − 1.19i)29-s + 0.143·31-s − 1.91·33-s + (0.443 + 0.443i)37-s + 1.01i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.110 - 0.993i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.378038601\)
\(L(\frac12)\) \(\approx\) \(1.378038601\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (1.66 - 1.66i)T - 3iT^{2} \)
7 \( 1 - 1.87iT - 7T^{2} \)
11 \( 1 + (-3.29 - 3.29i)T + 11iT^{2} \)
13 \( 1 + (-1.90 + 1.90i)T - 13iT^{2} \)
17 \( 1 - 2.57T + 17T^{2} \)
19 \( 1 + (-5.76 + 5.76i)T - 19iT^{2} \)
23 \( 1 - 7.58iT - 23T^{2} \)
29 \( 1 + (-6.45 + 6.45i)T - 29iT^{2} \)
31 \( 1 - 0.799T + 31T^{2} \)
37 \( 1 + (-2.69 - 2.69i)T + 37iT^{2} \)
41 \( 1 - 0.946iT - 41T^{2} \)
43 \( 1 + (0.829 + 0.829i)T + 43iT^{2} \)
47 \( 1 + 1.52T + 47T^{2} \)
53 \( 1 + (-6.97 - 6.97i)T + 53iT^{2} \)
59 \( 1 + (6.84 + 6.84i)T + 59iT^{2} \)
61 \( 1 + (6.87 - 6.87i)T - 61iT^{2} \)
67 \( 1 + (-3.73 + 3.73i)T - 67iT^{2} \)
71 \( 1 - 9.34iT - 71T^{2} \)
73 \( 1 + 0.886iT - 73T^{2} \)
79 \( 1 + 3.07T + 79T^{2} \)
83 \( 1 + (-0.989 + 0.989i)T - 83iT^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + 7.16T + 97T^{2} \)
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

−9.579848969248661807953345838444, −9.247610153911717075839417274226, −8.036761405125725983831458651373, −7.12274696940992561735442185347, −6.13291932845330798743943786217, −5.45925603042074075578361616523, −4.76480345159962519897689218759, −3.88962135081767423978711179291, −2.78029623756262779024007444155, −1.15028034845395104737637889627, 0.812595482325958065677987905483, 1.45075850886780049613113237678, 3.21858942015129556769072004453, 4.10953944338986652094301156600, 5.31922662020622159536685656312, 6.16036169060653594847200138817, 6.62520570845084823495778261724, 7.42499933143613336075183305398, 8.279390160403590794471449512818, 9.108072125894051832405273376288

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