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Properties

Label 2-40e2-20.3-c1-0-19
Degree $2$
Conductor $1600$
Sign $0.899 + 0.437i$
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  + (−1.58 + 1.58i)3-s − 2.00i·9-s − 3.87i·11-s + (2.44 + 2.44i)13-s + (1.22 − 1.22i)17-s − 3.87·19-s + (−3.16 + 3.16i)23-s + (−1.58 − 1.58i)27-s − 6i·29-s − 7.74i·31-s + (6.12 + 6.12i)33-s + (4.89 − 4.89i)37-s − 7.74·39-s − 3·41-s + (−3.16 − 3.16i)47-s + ⋯
L(s)  = 1  + (−0.912 + 0.912i)3-s − 0.666i·9-s − 1.16i·11-s + (0.679 + 0.679i)13-s + (0.297 − 0.297i)17-s − 0.888·19-s + (−0.659 + 0.659i)23-s + (−0.304 − 0.304i)27-s − 1.11i·29-s − 1.39i·31-s + (1.06 + 1.06i)33-s + (0.805 − 0.805i)37-s − 1.24·39-s − 0.468·41-s + (−0.461 − 0.461i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.899 + 0.437i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9507837592\)
\(L(\frac12)\) \(\approx\) \(0.9507837592\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 1.58i)T - 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + 3.87iT - 11T^{2} \)
13 \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \)
17 \( 1 + (-1.22 + 1.22i)T - 17iT^{2} \)
19 \( 1 + 3.87T + 19T^{2} \)
23 \( 1 + (3.16 - 3.16i)T - 23iT^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (3.16 + 3.16i)T + 47iT^{2} \)
53 \( 1 + (-2.44 - 2.44i)T + 53iT^{2} \)
59 \( 1 - 7.74T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-4.74 - 4.74i)T + 67iT^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 + (-3.67 - 3.67i)T + 73iT^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (-1.58 + 1.58i)T - 83iT^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (4.89 - 4.89i)T - 97iT^{2} \)
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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

−9.584888220320887944785232755989, −8.596532186690438567780746738489, −7.913588558569331924712568457687, −6.65557999218852686838288113934, −5.89765720788274216460214100131, −5.40174168050843737549319124604, −4.18526218871367109402218173134, −3.76443509031959078104109978565, −2.23774589387717210784102468451, −0.49562332881633218258206442361, 1.06452617722307528760107957366, 2.08775041315623824234372063847, 3.48311580131563074626154538058, 4.66603349457306597925222276492, 5.47662786775210001763851534472, 6.45142314319681176180384965334, 6.78832258253166973475215172765, 7.81148721414412585185284658366, 8.462480334501819080164444811268, 9.539138228491569276403519263786

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