Properties

Label 2-800-8.5-c1-0-3
Degree $2$
Conductor $800$
Sign $0.258 - 0.965i$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
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  + 0.732i·3-s − 2.73·7-s + 2.46·9-s − 2i·11-s + 3.46i·13-s + 3.46·17-s + 7.46i·19-s − 2i·21-s + 4.19·23-s + 4i·27-s + 6.92i·29-s − 1.46·31-s + 1.46·33-s − 2i·37-s − 2.53·39-s + ⋯
L(s)  = 1  + 0.422i·3-s − 1.03·7-s + 0.821·9-s − 0.603i·11-s + 0.960i·13-s + 0.840·17-s + 1.71i·19-s − 0.436i·21-s + 0.874·23-s + 0.769i·27-s + 1.28i·29-s − 0.262·31-s + 0.254·33-s − 0.328i·37-s − 0.406·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08374 + 0.831587i\)
\(L(\frac12)\) \(\approx\) \(1.08374 + 0.831587i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732iT - 3T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 7.46iT - 19T^{2} \)
23 \( 1 - 4.19T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 + 0.535iT - 59T^{2} \)
61 \( 1 + 4.92iT - 61T^{2} \)
67 \( 1 + 7.26iT - 67T^{2} \)
71 \( 1 - 1.46T + 71T^{2} \)
73 \( 1 + 0.535T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 4.73iT - 83T^{2} \)
89 \( 1 + 4.92T + 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−10.32805371524719015848111463676, −9.617945962840799651094781508613, −8.999369819261444313779531801457, −7.85050516282872322483477902373, −6.91084938234565956411098373044, −6.12047737831562567021336883993, −5.07441855356215912810014916301, −3.87471415567531540069732163171, −3.21917270886900169233480065245, −1.46472811326261188731775748312, 0.74503790584754474829739254472, 2.42079350331799264682391429705, 3.48443302444588795703526055651, 4.69914098317013200964109716396, 5.73713869313813131291205159471, 6.87538798215349305034241455904, 7.24268959836692863489301134774, 8.332874592748855555366033911579, 9.440954098168652739442408428681, 9.977327795166027493730404044021

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