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Properties

Label 1-847-847.129-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.924 + 0.380i$
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  + (−0.640 + 0.768i)2-s + (−0.913 − 0.406i)3-s + (−0.179 − 0.983i)4-s + (0.595 − 0.803i)5-s + (0.897 − 0.441i)6-s + (0.870 + 0.491i)8-s + (0.669 + 0.743i)9-s + (0.235 + 0.971i)10-s + (−0.235 + 0.971i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 + 0.491i)15-s + (−0.935 + 0.353i)16-s + (−0.999 − 0.0190i)17-s + (−0.999 + 0.0380i)18-s + (0.820 + 0.572i)19-s + (−0.897 − 0.441i)20-s + ⋯
L(s)  = 1  + (−0.640 + 0.768i)2-s + (−0.913 − 0.406i)3-s + (−0.179 − 0.983i)4-s + (0.595 − 0.803i)5-s + (0.897 − 0.441i)6-s + (0.870 + 0.491i)8-s + (0.669 + 0.743i)9-s + (0.235 + 0.971i)10-s + (−0.235 + 0.971i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 + 0.491i)15-s + (−0.935 + 0.353i)16-s + (−0.999 − 0.0190i)17-s + (−0.999 + 0.0380i)18-s + (0.820 + 0.572i)19-s + (−0.897 − 0.441i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.924 + 0.380i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.924 + 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04304880320 + 0.2180009483i\)
\(L(\frac12)\) \(\approx\) \(0.04304880320 + 0.2180009483i\)
\(L(1)\) \(\approx\) \(0.5106905279 + 0.09557998754i\)
\(L(1)\) \(\approx\) \(0.5106905279 + 0.09557998754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.640 + 0.768i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (0.595 - 0.803i)T \)
13 \( 1 + (-0.0285 + 0.999i)T \)
17 \( 1 + (-0.999 - 0.0190i)T \)
19 \( 1 + (0.820 + 0.572i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (-0.997 - 0.0760i)T \)
37 \( 1 + (-0.991 + 0.132i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.999 - 0.0380i)T \)
53 \( 1 + (-0.935 - 0.353i)T \)
59 \( 1 + (0.969 + 0.244i)T \)
61 \( 1 + (0.345 - 0.938i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (0.449 + 0.893i)T \)
79 \( 1 + (-0.953 + 0.299i)T \)
83 \( 1 + (-0.254 + 0.967i)T \)
89 \( 1 + (-0.981 + 0.189i)T \)
97 \( 1 + (-0.993 + 0.113i)T \)
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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

−21.86059142314479134593316764570, −20.94294989586298195421807046840, −20.311569755358758912101427532549, −19.27724736258248785317906892950, −18.27174674120414560384072504252, −17.88891651969943152169023040097, −17.299164448982822584669577187139, −16.34193237963749245372435790641, −15.53313292452988058146124551163, −14.55546236521391530932344056671, −13.29260030093077693136732883987, −12.74651069491596590510113882813, −11.6237344835664472626854543690, −10.96339413926917698081453711831, −10.43526753809505783642291527608, −9.638229396196335319646556548309, −8.873131068687815730556425383866, −7.51140535798551512886313786326, −6.7856758769583934304331844700, −5.74537907790338946636230160431, −4.74546174563069652258768579350, −3.60619803175014722443497224072, −2.72248734816642955969348345889, −1.578005775573391078897118902880, −0.143048455971673664061887152326, 1.395431490625941611055054848212, 1.909914343025141666101222543298, 4.14452890269201504218962508480, 5.16216444680440534426988391981, 5.66087831914640970344654739231, 6.637540865171839705031174145244, 7.3128847475552597405854295778, 8.36376112593312920298472615190, 9.3166082553299976360236063887, 9.88471162351865008621603679158, 11.03686694925233406192161744848, 11.70493810778900703970722902275, 12.877688799342273670416519726689, 13.55463433234449046463852812129, 14.36030873476172602092876751170, 15.63611035936204114373358950372, 16.310912129470160489383713531654, 16.85552864204301663408093348033, 17.619787223330891042850558696057, 18.1735247716091860025115934411, 19.01834891973850663720299666994, 19.866527784341077196068598913129, 20.821753377898893127417467886489, 21.88841642767342390403492695288, 22.537148867758883557164626228195

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