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Properties

Label 2-40e2-16.13-c1-0-20
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

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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: $\chi_{1600} (401, \cdot )$
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 \( 1 + 7.16T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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