Properties

Label 1-847-847.529-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.970 - 0.241i$
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  + (−0.786 − 0.618i)2-s + (−0.5 − 0.866i)3-s + (0.235 + 0.971i)4-s + (−0.995 − 0.0950i)5-s + (−0.142 + 0.989i)6-s + (0.415 − 0.909i)8-s + (−0.5 + 0.866i)9-s + (0.723 + 0.690i)10-s + (0.723 − 0.690i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)15-s + (−0.888 + 0.458i)16-s + (0.981 + 0.189i)17-s + (0.928 − 0.371i)18-s + (0.981 − 0.189i)19-s + (−0.142 − 0.989i)20-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)2-s + (−0.5 − 0.866i)3-s + (0.235 + 0.971i)4-s + (−0.995 − 0.0950i)5-s + (−0.142 + 0.989i)6-s + (0.415 − 0.909i)8-s + (−0.5 + 0.866i)9-s + (0.723 + 0.690i)10-s + (0.723 − 0.690i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)15-s + (−0.888 + 0.458i)16-s + (0.981 + 0.189i)17-s + (0.928 − 0.371i)18-s + (0.981 − 0.189i)19-s + (−0.142 − 0.989i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.970 - 0.241i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.970 - 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04802235620 - 0.3918974045i\)
\(L(\frac12)\) \(\approx\) \(0.04802235620 - 0.3918974045i\)
\(L(1)\) \(\approx\) \(0.4059035768 - 0.2635170694i\)
\(L(1)\) \(\approx\) \(0.4059035768 - 0.2635170694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.786 - 0.618i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.995 - 0.0950i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (0.981 + 0.189i)T \)
19 \( 1 + (0.981 - 0.189i)T \)
23 \( 1 + (-0.888 + 0.458i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (0.723 + 0.690i)T \)
37 \( 1 + (0.235 - 0.971i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (0.928 + 0.371i)T \)
53 \( 1 + (-0.888 - 0.458i)T \)
59 \( 1 + (-0.786 + 0.618i)T \)
61 \( 1 + (0.928 + 0.371i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.0475 - 0.998i)T \)
79 \( 1 + (-0.995 - 0.0950i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (-0.327 - 0.945i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−22.55955228085607691189274541513, −21.91271906786144500932309355405, −20.42859023957395515762422481994, −20.287559061541367727570339609525, −19.058679216497504889715732154207, −18.53426481235012644688921958456, −17.460262164047738815374715003918, −16.74808148332200577647385535308, −16.11936764533466409723918390495, −15.51895947802602422256166293158, −14.661816101309480058205829925970, −14.137145735175782771741827674289, −12.31254911821568664254236012890, −11.712032597218658007642010337626, −10.90953472368415548027975816858, −9.96925732205938738391354574383, −9.47159368153047028698002634657, −8.340081469868126134190154967451, −7.59107543325433767342317182500, −6.73272673982185753998524671137, −5.63736260527166694246897748847, −4.88215312016247471591063773602, −3.92459290521449646734910604256, −2.73125499767338032323642857179, −0.9755468907896979008417273777, 0.33395433634293568840208520877, 1.38111976414878395979622561211, 2.57087840310863082138253206725, 3.503563199869505106104884949522, 4.69560491663991376602710367380, 5.85317805800322181186398753211, 7.17361466815879605547323362174, 7.62770449757839036710108318494, 8.23747648524032654369034099223, 9.434115794479963460487777209327, 10.35877515583723837798681728152, 11.2706016373037032099033491812, 12.04920740900209704392365809467, 12.30041976955821258628095961518, 13.32029428984634194376659577461, 14.373336698781350809963728664211, 15.63647590310911811122626203775, 16.362471481534883718486889827507, 17.148832527388401094374416777592, 17.84756748759929758853769460058, 18.68090779727741976272774617611, 19.32626863243285001396691866302, 19.8522436618743912012109113559, 20.638090961480160302937672044061, 21.81674668470105930172481999319

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