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Properties

Label 1-847-847.640-r0-0-0
Degree $1$
Conductor $847$
Sign $0.309 + 0.950i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (640, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.886463152 + 2.096099568i\)
\(L(\frac12)\) \(\approx\) \(2.886463152 + 2.096099568i\)
\(L(1)\) \(\approx\) \(2.175790338 + 0.9113433843i\)
\(L(1)\) \(\approx\) \(2.175790338 + 0.9113433843i\)

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

−21.73995834382467657104749310459, −21.01108827943559451785963507402, −20.38124359694629182522793622924, −19.9922332595657321666701185995, −18.97504818746907573349331956326, −18.430075232602659309274498582520, −16.72312316371974958742488783347, −16.04384644084079352978688222015, −15.465239616980075691220268210, −14.56649853559294139809806721648, −13.65790659558945827435009809102, −13.29605557781073603201083883842, −12.25840444917814194498258604567, −11.54304959179708673978045680185, −10.6078008674169789469722691351, −9.28695680438453947945922326788, −9.12289552835008896700385541593, −7.67009387869217929674922212289, −7.07032011390529011500770566257, −5.56518197076990595466518193469, −4.8632330259043839572133090783, −3.84142232734553205927368250382, −3.36391528411108365872000245734, −2.051933395342853775155745395129, −1.185365553506017045709127653344, 1.65067710593824733746524057534, 2.85580417775304398397353161211, 3.39835709934057462439154842430, 4.16504681367013981748451692520, 5.449904904112149294634052784203, 6.52942552521886896760353263391, 7.18917027624745682258384019580, 8.01635692730169589955974235524, 8.659545316323695612803933678044, 10.05229327853955457607606964727, 10.96655318592843318628393782177, 11.8082196840718179817666439938, 12.95225124229245093214672709360, 13.32694598894646036452204024915, 14.41477770289810401131836710818, 14.88419857631348312784277090753, 15.497051639547738468737945630086, 16.29091498401305774616273212535, 17.49395217538602287941481284867, 18.42468729034221165392802730268, 19.10610449580158559258960976753, 20.26719651998039435452038737289, 20.45895586560043083403538220142, 21.71113995914282089381799783323, 22.195885221925033719728968016320

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