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Properties

Label 2-40e2-100.11-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6292167620\)
\(L(\frac12)\) \(\approx\) \(0.6292167620\)
\(L(1)\) 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

−9.676063369975670327358227614116, −8.470376387066196046239116137976, −7.26163154048967855301474884764, −7.08423195379189921771599361265, −6.08015600499275755730860184357, −5.41737684590961161715778816135, −4.42592180628927913427954571332, −3.30988391685286884074583068171, −2.07436226681351332262030798644, −0.52204646233237342968962369278, 1.88760119127643573341215305044, 2.63601913108886195616824733752, 4.37787100980143940373335555199, 5.16146807250773369423992958735, 5.65520188947811627115000272402, 6.29977771566192042382548132901, 7.35898576839707447428980619559, 8.669052122839764660360191335202, 9.025815959157027317747516900662, 9.838088196271625464117919703082

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