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Properties

Label 1-847-847.207-r0-0-0
Degree $1$
Conductor $847$
Sign $0.516 - 0.856i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (−0.905 − 0.424i)2-s + (−0.978 + 0.207i)3-s + (0.640 + 0.768i)4-s + (0.449 − 0.893i)5-s + (0.974 + 0.226i)6-s + (−0.254 − 0.967i)8-s + (0.913 − 0.406i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (−0.179 + 0.983i)16-s + (0.00951 + 0.999i)17-s + (−0.999 − 0.0190i)18-s + (0.953 − 0.299i)19-s + (0.974 − 0.226i)20-s + ⋯
L(s)  = 1  + (−0.905 − 0.424i)2-s + (−0.978 + 0.207i)3-s + (0.640 + 0.768i)4-s + (0.449 − 0.893i)5-s + (0.974 + 0.226i)6-s + (−0.254 − 0.967i)8-s + (0.913 − 0.406i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (−0.179 + 0.983i)16-s + (0.00951 + 0.999i)17-s + (−0.999 − 0.0190i)18-s + (0.953 − 0.299i)19-s + (0.974 − 0.226i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.516 - 0.856i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.516 - 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7199716971 - 0.4066803346i\)
\(L(\frac12)\) \(\approx\) \(0.7199716971 - 0.4066803346i\)
\(L(1)\) \(\approx\) \(0.6363767816 - 0.1913326036i\)
\(L(1)\) \(\approx\) \(0.6363767816 - 0.1913326036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.905 - 0.424i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (0.449 - 0.893i)T \)
13 \( 1 + (0.696 - 0.717i)T \)
17 \( 1 + (0.00951 + 0.999i)T \)
19 \( 1 + (0.953 - 0.299i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (0.993 + 0.113i)T \)
31 \( 1 + (0.999 - 0.0380i)T \)
37 \( 1 + (-0.0665 + 0.997i)T \)
41 \( 1 + (-0.921 + 0.389i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.999 + 0.0190i)T \)
53 \( 1 + (-0.179 - 0.983i)T \)
59 \( 1 + (0.123 + 0.992i)T \)
61 \( 1 + (0.820 + 0.572i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (-0.851 + 0.524i)T \)
79 \( 1 + (0.988 + 0.151i)T \)
83 \( 1 + (0.610 - 0.791i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (-0.998 - 0.0570i)T \)
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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

−22.47458462006538631019180231594, −21.38859732444335718328604351306, −20.74840428523875935999003386470, −19.3669073218981970828622911156, −18.85240948342509159220227181610, −18.00296627327045791607165445148, −17.726782579992833245383852579378, −16.71963191782407205498690924101, −15.98860777394384343118171711530, −15.35917363245651776105949244211, −14.10094729294623455334154446403, −13.63051319951847688572013564865, −12.10660172114480394065925118055, −11.35767976318189550773284012421, −10.81720434662167717262993376480, −9.86722717702013561644627100486, −9.28497722083751375943215887161, −7.91551207646482406564276433546, −7.07190377194313479266972991202, −6.51314908634835465543663585469, −5.71392528606277697072354451402, −4.82610425904666123025489478697, −3.1834121294660055405636721042, −1.9606082583537887920301331195, −0.96959469795578676437417033011, 0.82110696530715119649623668332, 1.439387340552523432651534321191, 2.918350703973749498505076785800, 4.12159552420718608544769169698, 5.12940150530554011241068621152, 6.09228502712370163059621900312, 6.8783770700613350173965039884, 8.1811168648284085706995411268, 8.75322991433109052417882736097, 9.92834292092710876532303514904, 10.29225321767549694192622702598, 11.30939032222343535307972796372, 12.04943223337440161498745447110, 12.824278843894131446115140360585, 13.4517284903637996291286043962, 15.13865562672611421265781960363, 15.96426972311946945959366004914, 16.50941629873128209865829765462, 17.4324027615489850324264389821, 17.7063772259317313670378454907, 18.63697139574115765874567576964, 19.5649600387194410784534206070, 20.575732863777195607672356366210, 20.97658413522406274634850732464, 21.83421201371513400561794337474

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