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Properties

Label 2-40e2-200.91-c0-0-1
Degree $2$
Conductor $1600$
Sign $0.790 - 0.612i$
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  + (0.190 + 0.587i)3-s + (0.951 − 0.309i)5-s + i·7-s + (0.5 − 0.363i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)13-s + (0.363 + 0.5i)15-s + (−0.190 + 0.587i)17-s + (0.5 − 1.53i)19-s + (−0.587 + 0.190i)21-s + (−0.587 + 0.809i)23-s + (0.809 − 0.587i)25-s + (0.809 + 0.587i)27-s + (−0.951 + 0.309i)29-s + (0.951 + 0.309i)31-s + ⋯
L(s)  = 1  + (0.190 + 0.587i)3-s + (0.951 − 0.309i)5-s + i·7-s + (0.5 − 0.363i)9-s + (−0.809 − 0.587i)11-s + (0.587 + 0.809i)13-s + (0.363 + 0.5i)15-s + (−0.190 + 0.587i)17-s + (0.5 − 1.53i)19-s + (−0.587 + 0.190i)21-s + (−0.587 + 0.809i)23-s + (0.809 − 0.587i)25-s + (0.809 + 0.587i)27-s + (−0.951 + 0.309i)29-s + (0.951 + 0.309i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.790 - 0.612i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.790 - 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.426532311\)
\(L(\frac12)\) \(\approx\) \(1.426532311\)
\(L(1)\) 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 \( 1 + (-0.951 + 0.309i)T \)
good3 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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

−9.541386620295537536318810021811, −8.937774770868036788140307875712, −8.537468221611841905818990739466, −7.19370930123109567343693515644, −6.30641281550387578135909283281, −5.51932703557317637118076168599, −4.86466514362880879952107586412, −3.72567002901169173172690439621, −2.68896384230871911619396135975, −1.62156375519937490333112924818, 1.34079449661293068961215765629, 2.29241112556012279011613663510, 3.40748246258609209332808353975, 4.57797405436112925732243640257, 5.50358742729721252183921799198, 6.39304697111008053672371014816, 7.21175406803272564326281592447, 7.78482597317664665804539529737, 8.528313620359183830583583858732, 9.920662966135597672838355891567

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