Properties

Label 2-800-8.5-c3-0-7
Degree $2$
Conductor $800$
Sign $0.353 - 0.935i$
Analytic cond. $47.2015$
Root an. cond. $6.87033$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.29i·3-s − 8·7-s − 1.00·9-s − 15.8i·11-s + 52.9i·13-s + 14·17-s − 37.0i·19-s + 42.3i·21-s − 152·23-s − 137. i·27-s + 158. i·29-s − 224·31-s − 84.0·33-s + 243. i·37-s + 280·39-s + ⋯
L(s)  = 1  − 1.01i·3-s − 0.431·7-s − 0.0370·9-s − 0.435i·11-s + 1.12i·13-s + 0.199·17-s − 0.447i·19-s + 0.439i·21-s − 1.37·23-s − 0.980i·27-s + 1.01i·29-s − 1.29·31-s − 0.443·33-s + 1.08i·37-s + 1.14·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.353 - 0.935i$
Analytic conductor: \(47.2015\)
Root analytic conductor: \(6.87033\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :3/2),\ 0.353 - 0.935i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9017986903\)
\(L(\frac12)\) \(\approx\) \(0.9017986903\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.29iT - 27T^{2} \)
7 \( 1 + 8T + 343T^{2} \)
11 \( 1 + 15.8iT - 1.33e3T^{2} \)
13 \( 1 - 52.9iT - 2.19e3T^{2} \)
17 \( 1 - 14T + 4.91e3T^{2} \)
19 \( 1 + 37.0iT - 6.85e3T^{2} \)
23 \( 1 + 152T + 1.21e4T^{2} \)
29 \( 1 - 158. iT - 2.43e4T^{2} \)
31 \( 1 + 224T + 2.97e4T^{2} \)
37 \( 1 - 243. iT - 5.06e4T^{2} \)
41 \( 1 + 70T + 6.89e4T^{2} \)
43 \( 1 - 439. iT - 7.95e4T^{2} \)
47 \( 1 - 336T + 1.03e5T^{2} \)
53 \( 1 - 31.7iT - 1.48e5T^{2} \)
59 \( 1 - 534. iT - 2.05e5T^{2} \)
61 \( 1 + 95.2iT - 2.26e5T^{2} \)
67 \( 1 + 174. iT - 3.00e5T^{2} \)
71 \( 1 - 72T + 3.57e5T^{2} \)
73 \( 1 - 294T + 3.89e5T^{2} \)
79 \( 1 - 464T + 4.93e5T^{2} \)
83 \( 1 - 545. iT - 5.71e5T^{2} \)
89 \( 1 - 266T + 7.04e5T^{2} \)
97 \( 1 + 994T + 9.12e5T^{2} \)
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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

−9.945180541227773339756653636036, −9.157821486564181833801476207821, −8.223625288263474519782614322533, −7.34953398701924564583947121570, −6.63231441976888985963369956135, −5.93434359289415222397581019190, −4.62120747926129853004260095382, −3.49601985253720186966975684082, −2.20252078332611003468659867997, −1.20275084940894231248047869013, 0.24853484667633110509320176956, 2.05035848570045410945505335531, 3.46771193590558800220132001357, 4.08533632653934119034433227853, 5.24718275168003991439486415249, 5.96289589358899119272046804422, 7.22270888911280730521447734003, 8.026227797089654370630206593256, 9.093052370770075259444537916697, 9.898476345477803808246889285268

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