Properties

Label 2-800-8.5-c5-0-37
Degree $2$
Conductor $800$
Sign $0.00874 - 0.999i$
Analytic cond. $128.307$
Root an. cond. $11.3272$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.7i·3-s + 198.·7-s + 127.·9-s − 85.9i·11-s − 407. i·13-s − 1.20e3·17-s + 206. i·19-s + 2.13e3i·21-s − 2.59e3·23-s + 3.98e3i·27-s + 6.19e3i·29-s + 1.86e3·31-s + 923.·33-s + 1.47e4i·37-s + 4.37e3·39-s + ⋯
L(s)  = 1  + 0.689i·3-s + 1.53·7-s + 0.524·9-s − 0.214i·11-s − 0.668i·13-s − 1.01·17-s + 0.130i·19-s + 1.05i·21-s − 1.02·23-s + 1.05i·27-s + 1.36i·29-s + 0.348·31-s + 0.147·33-s + 1.76i·37-s + 0.460·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00874 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.00874 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.00874 - 0.999i$
Analytic conductor: \(128.307\)
Root analytic conductor: \(11.3272\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :5/2),\ 0.00874 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.589610622\)
\(L(\frac12)\) \(\approx\) \(2.589610622\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.7iT - 243T^{2} \)
7 \( 1 - 198.T + 1.68e4T^{2} \)
11 \( 1 + 85.9iT - 1.61e5T^{2} \)
13 \( 1 + 407. iT - 3.71e5T^{2} \)
17 \( 1 + 1.20e3T + 1.41e6T^{2} \)
19 \( 1 - 206. iT - 2.47e6T^{2} \)
23 \( 1 + 2.59e3T + 6.43e6T^{2} \)
29 \( 1 - 6.19e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.86e3T + 2.86e7T^{2} \)
37 \( 1 - 1.47e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.80e4T + 1.15e8T^{2} \)
43 \( 1 + 9.26e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.43e4T + 2.29e8T^{2} \)
53 \( 1 - 1.27e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.07e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.13e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.26e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.12e4T + 1.80e9T^{2} \)
73 \( 1 + 2.32e4T + 2.07e9T^{2} \)
79 \( 1 + 2.91e4T + 3.07e9T^{2} \)
83 \( 1 - 4.80e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.01e4T + 5.58e9T^{2} \)
97 \( 1 - 1.13e5T + 8.58e9T^{2} \)
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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

−9.829055760924839840032225536085, −8.805998878001819002726515998249, −8.126612767766780811784137957575, −7.30192435106135733962752127582, −6.12846804399562366202695169181, −4.96777627182504535229515008111, −4.55062728153249575166220465507, −3.45505196578484690002281938467, −2.07626578319971181913713868359, −1.07930580398720528151792794064, 0.55134763466201568421025734177, 1.77897242583659381088160207347, 2.22312843097492048754441166216, 4.15293269760839087867100600976, 4.61901109420325951015323905405, 5.89794878421527816848210334329, 6.80634066049668857049592518242, 7.68988533834901448899621329340, 8.194924577800514045424429423428, 9.237843633260651198688089136736

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