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Properties

Label 2-800-40.27-c1-0-0
Degree $2$
Conductor $800$
Sign $-0.863 + 0.503i$
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  + (−1.22 + 1.22i)3-s + (−3.16 + 3.16i)7-s + 11-s + (3.16 + 3.16i)13-s + (−3.67 − 3.67i)17-s − 3i·19-s − 7.74i·21-s + (−3.67 − 3.67i)27-s − 7.74·29-s + (−1.22 + 1.22i)33-s + (−3.16 + 3.16i)37-s − 7.74·39-s − 41-s + (2.44 − 2.44i)43-s + (−3.16 + 3.16i)47-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (−1.19 + 1.19i)7-s + 0.301·11-s + (0.877 + 0.877i)13-s + (−0.891 − 0.891i)17-s − 0.688i·19-s − 1.69i·21-s + (−0.707 − 0.707i)27-s − 1.43·29-s + (−0.213 + 0.213i)33-s + (−0.519 + 0.519i)37-s − 1.24·39-s − 0.156·41-s + (0.373 − 0.373i)43-s + (−0.461 + 0.461i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.863 + 0.503i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ -0.863 + 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0853074 - 0.315646i\)
\(L(\frac12)\) \(\approx\) \(0.0853074 - 0.315646i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.22i)T - 3iT^{2} \)
7 \( 1 + (3.16 - 3.16i)T - 7iT^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (-3.16 - 3.16i)T + 13iT^{2} \)
17 \( 1 + (3.67 + 3.67i)T + 17iT^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 7.74T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (3.16 - 3.16i)T - 37iT^{2} \)
41 \( 1 + T + 41T^{2} \)
43 \( 1 + (-2.44 + 2.44i)T - 43iT^{2} \)
47 \( 1 + (3.16 - 3.16i)T - 47iT^{2} \)
53 \( 1 + (6.32 + 6.32i)T + 53iT^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 7.74iT - 61T^{2} \)
67 \( 1 + (-3.67 - 3.67i)T + 67iT^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 + (-1.22 + 1.22i)T - 73iT^{2} \)
79 \( 1 + 7.74T + 79T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - 83iT^{2} \)
89 \( 1 - 13iT - 89T^{2} \)
97 \( 1 + (4.89 + 4.89i)T + 97iT^{2} \)
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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

−10.94286621834576334397119293791, −9.701358777548585375599793288209, −9.296285711738682374711840297264, −8.535746435097053724187635920469, −7.01663001071608885537759008222, −6.28745353931274960159339195096, −5.50206470425297208538404586999, −4.56255692938041364309247116064, −3.45714594266378735630582257357, −2.20760173468704494485445778690, 0.17895965083878145492100681923, 1.47404084694181229385956359412, 3.39664657143676848463732176565, 4.04675724216499875093695814488, 5.70456978898989720541293880572, 6.30732719506645008106507993210, 6.99132059566382897436788642683, 7.83798643399826572423143728307, 8.972557054277815826941268537723, 9.914176746314285408209813588017

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