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Properties

Label 2-800-25.9-c1-0-12
Degree $2$
Conductor $800$
Sign $0.883 - 0.468i$
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  + (2.21 − 0.333i)5-s + (−2.42 + 1.76i)9-s + (2.62 + 3.61i)13-s + (5.03 + 1.63i)17-s + (4.77 − 1.47i)25-s + (1.77 + 5.45i)29-s + (−5.01 − 6.90i)37-s + (8.65 − 6.28i)41-s + (−4.77 + 4.70i)45-s + 7·49-s + (−11.4 + 3.73i)53-s + (6.73 + 4.89i)61-s + (7.01 + 7.12i)65-s + (−0.447 + 0.616i)73-s + (2.78 − 8.55i)81-s + ⋯
L(s)  = 1  + (0.988 − 0.148i)5-s + (−0.809 + 0.587i)9-s + (0.728 + 1.00i)13-s + (1.22 + 0.396i)17-s + (0.955 − 0.294i)25-s + (0.329 + 1.01i)29-s + (−0.824 − 1.13i)37-s + (1.35 − 0.982i)41-s + (−0.712 + 0.701i)45-s + 49-s + (−1.57 + 0.512i)53-s + (0.862 + 0.626i)61-s + (0.870 + 0.883i)65-s + (−0.0524 + 0.0721i)73-s + (0.309 − 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75981 + 0.437632i\)
\(L(\frac12)\) \(\approx\) \(1.75981 + 0.437632i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.333i)T \)
good3 \( 1 + (2.42 - 1.76i)T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.62 - 3.61i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.03 - 1.63i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.77 - 5.45i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.01 + 6.90i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.65 + 6.28i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (11.4 - 3.73i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-6.73 - 4.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.447 - 0.616i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-14.1 - 10.2i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.90 - 1.26i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.47883592769720468284431259667, −9.322611006371804317246043978537, −8.828675842109879088297701993484, −7.84755858828802554530273061719, −6.74580847845025810388934585066, −5.80915911267105538015272359216, −5.22639925756812934028126382754, −3.89029327914601095722313851634, −2.61432559501383260608576636747, −1.45682696267584176150521325341, 1.04918004465847250553674041552, 2.67507874919984967645003872521, 3.48748800990758407171795152784, 5.06371988852954688184914017899, 5.87650856240789132888248195746, 6.44563476077526137618003865627, 7.74683143624205338658771900243, 8.530151606627751658830445666802, 9.483601964938143959116035540650, 10.09221441783985027525874687301

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