Properties

Label 2-800-1.1-c3-0-0
Degree $2$
Conductor $800$
Sign $1$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.30·3-s − 28.3·7-s − 8.49·9-s − 65.2·11-s + 33.6·13-s − 73.3·17-s − 134.·19-s + 121.·21-s + 14.7·23-s + 152.·27-s − 224.·29-s − 68.8·31-s + 280.·33-s + 196.·37-s − 144.·39-s − 143.·41-s − 15.0·43-s − 134.·47-s + 458.·49-s + 315.·51-s − 262.·53-s + 576.·57-s + 119.·59-s + 16.5·61-s + 240.·63-s − 545.·67-s − 63.2·69-s + ⋯
L(s)  = 1  − 0.827·3-s − 1.52·7-s − 0.314·9-s − 1.78·11-s + 0.718·13-s − 1.04·17-s − 1.61·19-s + 1.26·21-s + 0.133·23-s + 1.08·27-s − 1.43·29-s − 0.398·31-s + 1.48·33-s + 0.872·37-s − 0.594·39-s − 0.545·41-s − 0.0534·43-s − 0.417·47-s + 1.33·49-s + 0.865·51-s − 0.681·53-s + 1.34·57-s + 0.264·59-s + 0.0347·61-s + 0.481·63-s − 0.994·67-s − 0.110·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1162231436\)
\(L(\frac12)\) \(\approx\) \(0.1162231436\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.30T + 27T^{2} \)
7 \( 1 + 28.3T + 343T^{2} \)
11 \( 1 + 65.2T + 1.33e3T^{2} \)
13 \( 1 - 33.6T + 2.19e3T^{2} \)
17 \( 1 + 73.3T + 4.91e3T^{2} \)
19 \( 1 + 134.T + 6.85e3T^{2} \)
23 \( 1 - 14.7T + 1.21e4T^{2} \)
29 \( 1 + 224.T + 2.43e4T^{2} \)
31 \( 1 + 68.8T + 2.97e4T^{2} \)
37 \( 1 - 196.T + 5.06e4T^{2} \)
41 \( 1 + 143.T + 6.89e4T^{2} \)
43 \( 1 + 15.0T + 7.95e4T^{2} \)
47 \( 1 + 134.T + 1.03e5T^{2} \)
53 \( 1 + 262.T + 1.48e5T^{2} \)
59 \( 1 - 119.T + 2.05e5T^{2} \)
61 \( 1 - 16.5T + 2.26e5T^{2} \)
67 \( 1 + 545.T + 3.00e5T^{2} \)
71 \( 1 - 199.T + 3.57e5T^{2} \)
73 \( 1 + 43.2T + 3.89e5T^{2} \)
79 \( 1 + 438.T + 4.93e5T^{2} \)
83 \( 1 + 1.22e3T + 5.71e5T^{2} \)
89 \( 1 - 723.T + 7.04e5T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−10.07061360332656488712451103688, −9.047166849529295143536432118449, −8.246369482434406539630248119332, −7.05685660387888095243589733146, −6.21089296966563334438438567013, −5.69441452532037748975409747220, −4.55819496858862933857354727117, −3.30970310313647368586941338858, −2.30255122758277711542873347029, −0.18231366634640298327123630803, 0.18231366634640298327123630803, 2.30255122758277711542873347029, 3.30970310313647368586941338858, 4.55819496858862933857354727117, 5.69441452532037748975409747220, 6.21089296966563334438438567013, 7.05685660387888095243589733146, 8.246369482434406539630248119332, 9.047166849529295143536432118449, 10.07061360332656488712451103688

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