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Properties

Label 2-40e2-100.11-c0-0-0
Degree $2$
Conductor $1600$
Sign $0.481 - 0.876i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.309i)3-s + (0.809 + 0.587i)5-s + 1.61i·7-s + (−1.30 − 0.951i)13-s + (0.587 + 0.809i)15-s + (0.587 − 0.190i)19-s + (−0.500 + 1.53i)21-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (−0.587 − 0.809i)27-s + (0.190 − 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.951 + 1.30i)35-s + (−0.809 − 0.587i)37-s + (−0.951 − 1.30i)39-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s + (0.809 + 0.587i)5-s + 1.61i·7-s + (−1.30 − 0.951i)13-s + (0.587 + 0.809i)15-s + (0.587 − 0.190i)19-s + (−0.500 + 1.53i)21-s + (0.587 + 0.809i)23-s + (0.309 + 0.951i)25-s + (−0.587 − 0.809i)27-s + (0.190 − 0.587i)29-s + (0.951 − 0.309i)31-s + (−0.951 + 1.30i)35-s + (−0.809 − 0.587i)37-s + (−0.951 − 1.30i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.619445058\)
\(L(\frac12)\) \(\approx\) \(1.619445058\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 - 1.61iT - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.767915790435049487126899058822, −9.026746583349750362584854540812, −8.314041409853013985340704189132, −7.49603923018831085658137580245, −6.43308297269872470469067160197, −5.55589910677853772032772435513, −5.01716769130026702365342631665, −3.34959809166844992203511801430, −2.75423087390888016174445077077, −2.11256992270369490942184708269, 1.26566049461063558131206863871, 2.32396081362791528953595427763, 3.35624735203707789921332226799, 4.54097468679055984725949787824, 5.08755749249675015523114409608, 6.54042787077993968078574410837, 7.11847987121376466758030389193, 7.894452539482400476278479003558, 8.686660838052442311950962629477, 9.448433221593847494321221253892

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