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Properties

Label 2-800-40.19-c4-0-35
Degree $2$
Conductor $800$
Sign $0.0519 - 0.998i$
Analytic cond. $82.6959$
Root an. cond. $9.09373$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.9i·3-s − 56.7·7-s − 173.·9-s + 122.·11-s + 77.6·13-s − 196. i·17-s + 400.·19-s − 904. i·21-s + 806.·23-s − 1.46e3i·27-s − 147. i·29-s − 1.05e3i·31-s + 1.95e3i·33-s + 1.21e3·37-s + 1.23e3i·39-s + ⋯
L(s)  = 1  + 1.77i·3-s − 1.15·7-s − 2.13·9-s + 1.01·11-s + 0.459·13-s − 0.679i·17-s + 1.10·19-s − 2.05i·21-s + 1.52·23-s − 2.01i·27-s − 0.175i·29-s − 1.09i·31-s + 1.79i·33-s + 0.885·37-s + 0.813i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0519 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.0519 - 0.998i$
Analytic conductor: \(82.6959\)
Root analytic conductor: \(9.09373\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :2),\ 0.0519 - 0.998i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.965774237\)
\(L(\frac12)\) \(\approx\) \(1.965774237\)
\(L(3)\) 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 \( 1 - 15.9iT - 81T^{2} \)
7 \( 1 + 56.7T + 2.40e3T^{2} \)
11 \( 1 - 122.T + 1.46e4T^{2} \)
13 \( 1 - 77.6T + 2.85e4T^{2} \)
17 \( 1 + 196. iT - 8.35e4T^{2} \)
19 \( 1 - 400.T + 1.30e5T^{2} \)
23 \( 1 - 806.T + 2.79e5T^{2} \)
29 \( 1 + 147. iT - 7.07e5T^{2} \)
31 \( 1 + 1.05e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.21e3T + 1.87e6T^{2} \)
41 \( 1 + 2.41e3T + 2.82e6T^{2} \)
43 \( 1 - 922. iT - 3.41e6T^{2} \)
47 \( 1 - 3.72e3T + 4.87e6T^{2} \)
53 \( 1 - 3.43e3T + 7.89e6T^{2} \)
59 \( 1 + 1.90e3T + 1.21e7T^{2} \)
61 \( 1 - 700. iT - 1.38e7T^{2} \)
67 \( 1 - 1.05e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.07e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.12e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.44e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.58e3iT - 4.74e7T^{2} \)
89 \( 1 - 4.84e3T + 6.27e7T^{2} \)
97 \( 1 - 2.61e3iT - 8.85e7T^{2} \)
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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

−9.600007818218926735489567862743, −9.435732817109693209595857949483, −8.650342838545689870819327010114, −7.22736158863225519239791601818, −6.20482404682867547078932217045, −5.35104744859431469046530081808, −4.35611005889812264655436047232, −3.52487854360384978000395147028, −2.86312447466292165706374029983, −0.72079954930435192913256239541, 0.74287211840348642794745126136, 1.46622860725953098769739139061, 2.80343957602274318478193069575, 3.63683018786208237002460249336, 5.39610590763646963095493769161, 6.33378526995271510400770432637, 6.82773254586713829431468028952, 7.50596110406468320074373210503, 8.661726742263366032088985897512, 9.166839975101157310206772355603

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