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Properties

Label 2-800-5.3-c2-0-31
Degree $2$
Conductor $800$
Sign $0.130 + 0.991i$
Analytic cond. $21.7984$
Root an. cond. $4.66887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 + 1.54i)3-s + (2 − 2i)7-s − 4.19i·9-s − 1.10·11-s + (−10.0 − 10.0i)13-s + (−3.91 + 3.91i)17-s − 23.3i·19-s + 6.19·21-s + (−5.29 − 5.29i)23-s + (20.4 − 20.4i)27-s − 32.5i·29-s − 42.3·31-s + (−1.70 − 1.70i)33-s + (−22.2 + 22.2i)37-s − 31.1i·39-s + ⋯
L(s)  = 1  + (0.516 + 0.516i)3-s + (0.285 − 0.285i)7-s − 0.466i·9-s − 0.100·11-s + (−0.772 − 0.772i)13-s + (−0.230 + 0.230i)17-s − 1.23i·19-s + 0.295·21-s + (−0.230 − 0.230i)23-s + (0.757 − 0.757i)27-s − 1.12i·29-s − 1.36·31-s + (−0.0518 − 0.0518i)33-s + (−0.602 + 0.602i)37-s − 0.797i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.130 + 0.991i$
Analytic conductor: \(21.7984\)
Root analytic conductor: \(4.66887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1),\ 0.130 + 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.605228658\)
\(L(\frac12)\) \(\approx\) \(1.605228658\)
\(L(2)\) 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 \( 1 + (-1.54 - 1.54i)T + 9iT^{2} \)
7 \( 1 + (-2 + 2i)T - 49iT^{2} \)
11 \( 1 + 1.10T + 121T^{2} \)
13 \( 1 + (10.0 + 10.0i)T + 169iT^{2} \)
17 \( 1 + (3.91 - 3.91i)T - 289iT^{2} \)
19 \( 1 + 23.3iT - 361T^{2} \)
23 \( 1 + (5.29 + 5.29i)T + 529iT^{2} \)
29 \( 1 + 32.5iT - 841T^{2} \)
31 \( 1 + 42.3T + 961T^{2} \)
37 \( 1 + (22.2 - 22.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 15T + 1.68e3T^{2} \)
43 \( 1 + (-32.5 - 32.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-55.2 + 55.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (16.6 + 16.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 111. iT - 3.48e3T^{2} \)
61 \( 1 - 5.40T + 3.72e3T^{2} \)
67 \( 1 + (-36.0 + 36.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 71.2T + 5.04e3T^{2} \)
73 \( 1 + (-93.0 - 93.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 118. iT - 6.24e3T^{2} \)
83 \( 1 + (-6.64 - 6.64i)T + 6.88e3iT^{2} \)
89 \( 1 - 126. iT - 7.92e3T^{2} \)
97 \( 1 + (-73.4 + 73.4i)T - 9.40e3iT^{2} \)
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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

−9.795335211540416197006605538472, −9.117760665930688620730738444755, −8.245381059390656935350543941761, −7.39362459079578174260720989083, −6.43231129009266935587379104870, −5.24722602517486059033427119099, −4.35305363253888585211965728376, −3.35940517983085051601561789155, −2.30012610319989244549339433936, −0.48725259127449459632563834249, 1.62921792043336268037728475492, 2.43980582855695764931656499607, 3.76311808565655129789417224798, 4.94798096790796964658716784308, 5.81372743008433542495962213225, 7.15662018143369750157651818137, 7.53880371349941586259405808014, 8.608435260180344411460731328575, 9.195476095605219739560669922870, 10.29909243237799362146040351957

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