Properties

Label 2-800-100.23-c1-0-7
Degree $2$
Conductor $800$
Sign $0.578 - 0.815i$
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  + (−0.0719 − 0.141i)3-s + (−0.475 + 2.18i)5-s + (2.98 + 2.98i)7-s + (1.74 − 2.40i)9-s + (−1.53 − 2.10i)11-s + (−0.442 − 2.79i)13-s + (0.342 − 0.0899i)15-s + (5.14 + 2.62i)17-s + (0.539 + 1.66i)19-s + (0.206 − 0.636i)21-s + (−1.32 + 8.39i)23-s + (−4.54 − 2.07i)25-s + (−0.934 − 0.148i)27-s + (7.60 + 2.46i)29-s + (−4.78 + 1.55i)31-s + ⋯
L(s)  = 1  + (−0.0415 − 0.0815i)3-s + (−0.212 + 0.977i)5-s + (1.12 + 1.12i)7-s + (0.582 − 0.802i)9-s + (−0.461 − 0.635i)11-s + (−0.122 − 0.774i)13-s + (0.0884 − 0.0232i)15-s + (1.24 + 0.636i)17-s + (0.123 + 0.380i)19-s + (0.0451 − 0.138i)21-s + (−0.277 + 1.75i)23-s + (−0.909 − 0.415i)25-s + (−0.179 − 0.0284i)27-s + (1.41 + 0.458i)29-s + (−0.859 + 0.279i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.578 - 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49218 + 0.770675i\)
\(L(\frac12)\) \(\approx\) \(1.49218 + 0.770675i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.475 - 2.18i)T \)
good3 \( 1 + (0.0719 + 0.141i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (-2.98 - 2.98i)T + 7iT^{2} \)
11 \( 1 + (1.53 + 2.10i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.442 + 2.79i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-5.14 - 2.62i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.539 - 1.66i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.32 - 8.39i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-7.60 - 2.46i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.78 - 1.55i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.15 - 0.499i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-1.17 - 0.857i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (5.27 - 5.27i)T - 43iT^{2} \)
47 \( 1 + (-8.06 + 4.10i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-11.8 + 6.04i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (3.70 + 2.69i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.83 - 2.06i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.15 - 2.25i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-5.42 - 1.76i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.08 - 0.487i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (2.59 - 7.98i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.08 - 3.60i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (2.13 + 2.93i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.94 + 13.6i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.40496637702945993269675177629, −9.700076174309140837995727410028, −8.469631093166615239206222833909, −7.915875612956593785158785450665, −7.03025137213022585821344813085, −5.79814442411573820239714518179, −5.37011304510713199463936509792, −3.75824334175491500440177824204, −2.93416978179064432899213933695, −1.51070318063279452855115213539, 0.971658967863406395748747821300, 2.18594832987649236433296766745, 4.14348518142853187219566240746, 4.63727410897254519875706302736, 5.32253337414042607041744243629, 6.97135403196379713938052776974, 7.65492739730725490185424930983, 8.245482869718410388997456884223, 9.308248509855121620316802553763, 10.32838822507425973337885788135

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