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Properties

Label 2-40e2-80.43-c1-0-26
Degree $2$
Conductor $1600$
Sign $-0.826 + 0.562i$
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  − 1.39i·3-s + (−2.13 + 2.13i)7-s + 1.05·9-s + (−2.17 + 2.17i)11-s − 1.54·13-s + (3.86 − 3.86i)17-s + (0.0136 − 0.0136i)19-s + (2.97 + 2.97i)21-s + (−3.15 − 3.15i)23-s − 5.65i·27-s + (−3.33 − 3.33i)29-s − 8.92i·31-s + (3.02 + 3.02i)33-s − 7.24·37-s + 2.15i·39-s + ⋯
L(s)  = 1  − 0.804i·3-s + (−0.806 + 0.806i)7-s + 0.353·9-s + (−0.654 + 0.654i)11-s − 0.428·13-s + (0.937 − 0.937i)17-s + (0.00313 − 0.00313i)19-s + (0.648 + 0.648i)21-s + (−0.657 − 0.657i)23-s − 1.08i·27-s + (−0.619 − 0.619i)29-s − 1.60i·31-s + (0.526 + 0.526i)33-s − 1.19·37-s + 0.345i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7722790292\)
\(L(\frac12)\) \(\approx\) \(0.7722790292\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.39iT - 3T^{2} \)
7 \( 1 + (2.13 - 2.13i)T - 7iT^{2} \)
11 \( 1 + (2.17 - 2.17i)T - 11iT^{2} \)
13 \( 1 + 1.54T + 13T^{2} \)
17 \( 1 + (-3.86 + 3.86i)T - 17iT^{2} \)
19 \( 1 + (-0.0136 + 0.0136i)T - 19iT^{2} \)
23 \( 1 + (3.15 + 3.15i)T + 23iT^{2} \)
29 \( 1 + (3.33 + 3.33i)T + 29iT^{2} \)
31 \( 1 + 8.92iT - 31T^{2} \)
37 \( 1 + 7.24T + 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 2.02T + 43T^{2} \)
47 \( 1 + (3.34 + 3.34i)T + 47iT^{2} \)
53 \( 1 - 7.30iT - 53T^{2} \)
59 \( 1 + (3.52 + 3.52i)T + 59iT^{2} \)
61 \( 1 + (-1.41 + 1.41i)T - 61iT^{2} \)
67 \( 1 - 0.748T + 67T^{2} \)
71 \( 1 - 0.269T + 71T^{2} \)
73 \( 1 + (-0.811 + 0.811i)T - 73iT^{2} \)
79 \( 1 + 2.80T + 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (6.33 - 6.33i)T - 97iT^{2} \)
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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

−9.227617696604776970236306269412, −8.057508812903822301260535389873, −7.48298449834150514123192052848, −6.76825520733300670868851513983, −5.88824149023536636852776487365, −5.11787213125309911184302966184, −3.94398343507601315571244830622, −2.68464326682058879838215139678, −1.99733882144724421838382646298, −0.29296722827250779643333556890, 1.47023198439760154996746171153, 3.27382579167972147498940420714, 3.59636096507967960762874447329, 4.74237613077095774425209717415, 5.51973132370818857626730011631, 6.52571760528471283959692358308, 7.35270369328505010046474758263, 8.141057009766067319311510333823, 9.093557474860275259381200505104, 10.01991606562300845276015136074

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