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Properties

Label 1-805-805.558-r0-0-0
Degree $1$
Conductor $805$
Sign $-0.509 + 0.860i$
Analytic cond. $3.73840$
Root an. cond. $3.73840$
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  + (0.945 + 0.327i)2-s + (0.458 + 0.888i)3-s + (0.786 + 0.618i)4-s + (0.142 + 0.989i)6-s + (0.540 + 0.841i)8-s + (−0.580 + 0.814i)9-s + (−0.327 − 0.945i)11-s + (−0.189 + 0.981i)12-s + (0.281 + 0.959i)13-s + (0.235 + 0.971i)16-s + (−0.371 + 0.928i)17-s + (−0.814 + 0.580i)18-s + (0.928 − 0.371i)19-s − i·22-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.945 + 0.327i)2-s + (0.458 + 0.888i)3-s + (0.786 + 0.618i)4-s + (0.142 + 0.989i)6-s + (0.540 + 0.841i)8-s + (−0.580 + 0.814i)9-s + (−0.327 − 0.945i)11-s + (−0.189 + 0.981i)12-s + (0.281 + 0.959i)13-s + (0.235 + 0.971i)16-s + (−0.371 + 0.928i)17-s + (−0.814 + 0.580i)18-s + (0.928 − 0.371i)19-s − i·22-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $-0.509 + 0.860i$
Analytic conductor: \(3.73840\)
Root analytic conductor: \(3.73840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (558, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 805,\ (0:\ ),\ -0.509 + 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.474544175 + 2.587137787i\)
\(L(\frac12)\) \(\approx\) \(1.474544175 + 2.587137787i\)
\(L(1)\) \(\approx\) \(1.669718930 + 1.214103755i\)
\(L(1)\) \(\approx\) \(1.669718930 + 1.214103755i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.945 + 0.327i)T \)
3 \( 1 + (0.458 + 0.888i)T \)
11 \( 1 + (-0.327 - 0.945i)T \)
13 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (-0.371 + 0.928i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (-0.814 - 0.580i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.690 + 0.723i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (-0.189 - 0.981i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.618 + 0.786i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (0.909 - 0.415i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.27198197171660107877305352409, −20.79679101653981348345336357014, −20.54112239309864510824385191609, −19.8220912652445198309667058630, −18.90520030203668320250146934294, −18.12213875197691244484388529894, −17.37069839479231160040679602482, −15.92729999856242189527258401445, −15.35817015197252646935909321366, −14.47129426110407589037180619574, −13.65256604791862793518232601251, −13.14716654858887457167673099392, −12.17050836165930452146221485623, −11.77904101336337025680712807141, −10.51067914424979545446413426698, −9.7315580581768749600629190887, −8.51437582140943639433665983877, −7.39701250107683908151292053712, −6.94768215809394713821500671695, −5.76175872177122986334889864101, −5.025590237292510510322323981955, −3.74810816962470916591607030548, −2.86451951058759178297659626541, −2.06336525376237715667754763711, −0.93615398171171538807432886362, 1.84287412685975035176495635127, 2.98012069647410599817136409966, 3.6943103097508449438877889443, 4.550129331157190837676312777069, 5.44290412587604333657857333120, 6.27712798605283828959158257646, 7.40564370268222992405688366082, 8.39611312389970383188669512211, 9.05785614419051904344113345831, 10.32758195687207429726059926394, 11.13107216933234960399463492581, 11.76389857639935925528905908173, 13.08375998862344777048143051638, 13.6970266711918616584174689876, 14.4157672854736725573248902011, 15.175485479012281798064244239595, 16.03354887198719972977345308308, 16.452058573388691252073658479481, 17.33766731037415986999469963826, 18.65507833527152602387208815605, 19.64539749613129180610633045311, 20.31988881595817594755096069057, 21.2156231037062287198198106575, 21.7177761730806206090833568384, 22.25998490230279471530505096380

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