↑

Properties

Label 1-847-847.170-r0-0-0
Degree $1$
Conductor $847$
Sign $0.643 - 0.765i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.272 − 0.962i)2-s + (−0.978 − 0.207i)3-s + (−0.851 − 0.524i)4-s + (0.861 + 0.508i)5-s + (−0.466 + 0.884i)6-s + (−0.736 + 0.676i)8-s + (0.913 + 0.406i)9-s + (0.723 − 0.690i)10-s + (0.723 + 0.690i)12-s + (0.610 − 0.791i)13-s + (−0.736 − 0.676i)15-s + (0.449 + 0.893i)16-s + (−0.905 − 0.424i)17-s + (0.640 − 0.768i)18-s + (0.123 + 0.992i)19-s + (−0.466 − 0.884i)20-s + ⋯
L(s)  = 1  + (0.272 − 0.962i)2-s + (−0.978 − 0.207i)3-s + (−0.851 − 0.524i)4-s + (0.861 + 0.508i)5-s + (−0.466 + 0.884i)6-s + (−0.736 + 0.676i)8-s + (0.913 + 0.406i)9-s + (0.723 − 0.690i)10-s + (0.723 + 0.690i)12-s + (0.610 − 0.791i)13-s + (−0.736 − 0.676i)15-s + (0.449 + 0.893i)16-s + (−0.905 − 0.424i)17-s + (0.640 − 0.768i)18-s + (0.123 + 0.992i)19-s + (−0.466 − 0.884i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.643 - 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114164361 - 0.5192140353i\)
\(L(\frac12)\) \(\approx\) \(1.114164361 - 0.5192140353i\)
\(L(1)\) \(\approx\) \(0.8853781907 - 0.4073573034i\)
\(L(1)\) \(\approx\) \(0.8853781907 - 0.4073573034i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.272 - 0.962i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (0.861 + 0.508i)T \)
13 \( 1 + (0.610 - 0.791i)T \)
17 \( 1 + (-0.905 - 0.424i)T \)
19 \( 1 + (0.123 + 0.992i)T \)
23 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (0.516 + 0.856i)T \)
31 \( 1 + (-0.179 + 0.983i)T \)
37 \( 1 + (0.997 + 0.0760i)T \)
41 \( 1 + (0.897 - 0.441i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.640 + 0.768i)T \)
53 \( 1 + (0.449 - 0.893i)T \)
59 \( 1 + (-0.830 + 0.556i)T \)
61 \( 1 + (-0.969 - 0.244i)T \)
67 \( 1 + (0.928 + 0.371i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (0.964 + 0.263i)T \)
79 \( 1 + (0.749 - 0.662i)T \)
83 \( 1 + (-0.362 - 0.931i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (-0.870 - 0.491i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.184212150510900843108781210826, −21.64754695046106149264248571351, −21.117966113665479642119274790387, −19.89991162961870992136261015460, −18.49601568969803065249742795122, −18.00824421186943371568065297345, −17.143080637862321230271846553187, −16.79518975001094721356254421315, −15.78475153027767358489607242896, −15.342074332877208703145237617, −13.97985911519245208928076722386, −13.43812867833950664001733590802, −12.66426331461910195775655022964, −11.74084466884884880601825123251, −10.78521459212789828425789724636, −9.57830526817651014374734564676, −9.163305748017985827900126156929, −8.01871300510065353648735091383, −6.82853581077745521069214188744, −6.177237921305539994287206295547, −5.58247171685275009055024936147, −4.53113750994397098141670422028, −4.03141443459012244457676048955, −2.21640790638060995880088325864, −0.75855921319134356708961476900, 0.977237063323343152960887658790, 1.93924831280336403665872621008, 2.94631026389141225617801312266, 4.12393765866523731646046866403, 5.140152363542547181373518358226, 5.91843586759581123457701149829, 6.544307810221662718112183821035, 7.89424787028985502916819163319, 9.124361684370115556058666854706, 10.04420754160116567784140025384, 10.68570118744566179224963092830, 11.19596325697258969968314196436, 12.330667740877122593628723590387, 12.84669188790273206092210773166, 13.754572929584724716619046891368, 14.39956816909835204800485045952, 15.57760974179117782054137311075, 16.527061057799236588433772376423, 17.68831690383183092144851607954, 18.04924823409781669569584101250, 18.55904779290960732454028602519, 19.66799507648516370422979450370, 20.54599669415321484392944856743, 21.38290113352786969916074656939, 21.997911953077479074494107536108

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