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Properties

Label 2-800-100.31-c0-0-1
Degree $2$
Conductor $800$
Sign $0.995 + 0.0941i$
Analytic cond. $0.399252$
Root an. cond. $0.631863$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 0.809i)3-s + (0.309 − 0.951i)5-s − 0.618i·7-s + (0.190 − 0.587i)13-s + (0.951 − 0.309i)15-s + (−0.951 + 1.30i)19-s + (0.500 − 0.363i)21-s + (0.951 − 0.309i)23-s + (−0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (−1.30 + 0.951i)29-s + (−0.587 + 0.809i)31-s + (−0.587 − 0.190i)35-s + (−0.309 + 0.951i)37-s + (0.587 − 0.190i)39-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s + (0.309 − 0.951i)5-s − 0.618i·7-s + (0.190 − 0.587i)13-s + (0.951 − 0.309i)15-s + (−0.951 + 1.30i)19-s + (0.500 − 0.363i)21-s + (0.951 − 0.309i)23-s + (−0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (−1.30 + 0.951i)29-s + (−0.587 + 0.809i)31-s + (−0.587 − 0.190i)35-s + (−0.309 + 0.951i)37-s + (0.587 − 0.190i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.995 + 0.0941i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ 0.995 + 0.0941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217085292\)
\(L(\frac12)\) \(\approx\) \(1.217085292\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + 0.618iT - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−10.28198166550133449347622570320, −9.627997285487594393415620366078, −8.760581316940946647199114825535, −8.257766539249353292932676370082, −7.08352407393039407715291082121, −5.91850058211446209547036506391, −4.92057367600375059294397485505, −4.06290126220015714186479761468, −3.20307752340291635623029196937, −1.48160198417663858990829979556, 2.01518637056998855507874392963, 2.57856052430192880723067752895, 3.88833038792028134407616897199, 5.31622817989715229523872945707, 6.34426586490685552068014829226, 7.10803202294893138814817695815, 7.73452042945538029690784694539, 8.916106903831152728694037213943, 9.346156279804282727003416352525, 10.70008665261545129435102571691

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