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Properties

Label 2-800-100.23-c1-0-0
Degree $2$
Conductor $800$
Sign $0.495 - 0.868i$
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  + (−1.46 − 2.87i)3-s + (−1.81 + 1.30i)5-s + (−2.79 − 2.79i)7-s + (−4.35 + 5.99i)9-s + (−1.60 − 2.20i)11-s + (−0.689 − 4.35i)13-s + (6.40 + 3.32i)15-s + (4.55 + 2.32i)17-s + (0.743 + 2.28i)19-s + (−3.93 + 12.1i)21-s + (−0.877 + 5.54i)23-s + (1.61 − 4.73i)25-s + (14.0 + 2.22i)27-s + (−6.14 − 1.99i)29-s + (−3.04 + 0.989i)31-s + ⋯
L(s)  = 1  + (−0.845 − 1.65i)3-s + (−0.813 + 0.581i)5-s + (−1.05 − 1.05i)7-s + (−1.45 + 1.99i)9-s + (−0.483 − 0.665i)11-s + (−0.191 − 1.20i)13-s + (1.65 + 0.858i)15-s + (1.10 + 0.563i)17-s + (0.170 + 0.524i)19-s + (−0.859 + 2.64i)21-s + (−0.183 + 1.15i)23-s + (0.323 − 0.946i)25-s + (2.70 + 0.427i)27-s + (−1.14 − 0.370i)29-s + (−0.546 + 0.177i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.495 - 0.868i$
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: \((2,\ 800,\ (\ :1/2),\ 0.495 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0649026 + 0.0377204i\)
\(L(\frac12)\) \(\approx\) \(0.0649026 + 0.0377204i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 - 1.30i)T \)
good3 \( 1 + (1.46 + 2.87i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (2.79 + 2.79i)T + 7iT^{2} \)
11 \( 1 + (1.60 + 2.20i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.689 + 4.35i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.55 - 2.32i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.743 - 2.28i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.877 - 5.54i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (6.14 + 1.99i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.04 - 0.989i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.09 + 0.964i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-5.32 - 3.86i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.82 + 2.82i)T - 43iT^{2} \)
47 \( 1 + (-1.10 + 0.561i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (4.64 - 2.36i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.214 + 0.156i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.418 + 0.304i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.64 - 5.18i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (2.21 + 0.719i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.4 + 1.65i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (3.73 - 11.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.80 - 2.95i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-7.20 - 9.91i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.94 + 9.70i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.71048576110274139823277045576, −9.842842659803240649773727617442, −8.073822732117983371166859547445, −7.64469572856830016335227780111, −7.15501852277355196047545247548, −6.03618921734877263039345128068, −5.61047028450593859485525990367, −3.74407438197458161140567726901, −2.86798254071294866585702430262, −1.04138720615890412091553286568, 0.05066360828354214149316243397, 2.83642523563137708887137737179, 3.92001802226192183876430517056, 4.71649241477784898070623221005, 5.45193488073618512930873093869, 6.33861005805543457298102637737, 7.54586281247283037638876467500, 9.053010629545314229043815101093, 9.221669120882711162700829777087, 9.993456601657315852577973630366

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