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Properties

Label 1-847-847.831-r0-0-0
Degree $1$
Conductor $847$
Sign $0.538 + 0.842i$
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  + (0.290 + 0.956i)2-s + (0.978 + 0.207i)3-s + (−0.830 + 0.556i)4-s + (0.625 − 0.780i)5-s + (0.0855 + 0.996i)6-s + (−0.774 − 0.633i)8-s + (0.913 + 0.406i)9-s + (0.928 + 0.371i)10-s + (−0.928 + 0.371i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (0.380 − 0.924i)16-s + (0.749 + 0.662i)17-s + (−0.123 + 0.992i)18-s + (−0.398 − 0.917i)19-s + (−0.0855 + 0.996i)20-s + ⋯
L(s)  = 1  + (0.290 + 0.956i)2-s + (0.978 + 0.207i)3-s + (−0.830 + 0.556i)4-s + (0.625 − 0.780i)5-s + (0.0855 + 0.996i)6-s + (−0.774 − 0.633i)8-s + (0.913 + 0.406i)9-s + (0.928 + 0.371i)10-s + (−0.928 + 0.371i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (0.380 − 0.924i)16-s + (0.749 + 0.662i)17-s + (−0.123 + 0.992i)18-s + (−0.398 − 0.917i)19-s + (−0.0855 + 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.538 + 0.842i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.538 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.212739544 + 1.212656550i\)
\(L(\frac12)\) \(\approx\) \(2.212739544 + 1.212656550i\)
\(L(1)\) \(\approx\) \(1.583901395 + 0.7045434896i\)
\(L(1)\) \(\approx\) \(1.583901395 + 0.7045434896i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.290 + 0.956i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.625 - 0.780i)T \)
13 \( 1 + (-0.466 - 0.884i)T \)
17 \( 1 + (0.749 + 0.662i)T \)
19 \( 1 + (-0.398 - 0.917i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (0.736 + 0.676i)T \)
31 \( 1 + (0.969 - 0.244i)T \)
37 \( 1 + (0.345 + 0.938i)T \)
41 \( 1 + (0.516 - 0.856i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.123 - 0.992i)T \)
53 \( 1 + (0.380 + 0.924i)T \)
59 \( 1 + (0.999 - 0.0190i)T \)
61 \( 1 + (-0.683 - 0.730i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (-0.432 - 0.901i)T \)
79 \( 1 + (-0.548 - 0.836i)T \)
83 \( 1 + (0.941 + 0.336i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (0.362 - 0.931i)T \)
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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

−21.62158857551932826406159592880, −21.17812526942296948012999194495, −20.5680336143636249706132317498, −19.45742661915504148149238295586, −18.98424879160807421336551242390, −18.37527019427347850677114008381, −17.55512748025703583681189267920, −16.3504619111695331595282229657, −15.03573343578382709546878005060, −14.356572475811404501577174064046, −14.04736162385873574456668725400, −13.12196260887551436219049815125, −12.28442217130438615840987357691, −11.448752372593414839148447172371, −10.213790117140019908314588279847, −9.895095171604703131005808839091, −8.98855959492797288107813173908, −8.04122129002026001379492049986, −6.901255096343990111927074423053, −6.03301354047088875354066775125, −4.73528186932516327783888493325, −3.83149692138726378124872717870, −2.7969499251540875889955823839, −2.29867714005614201056786892297, −1.24502769343068119940470576528, 1.13329551943762470966676459839, 2.58703591985323897706426357469, 3.52666091488035331296235960440, 4.64342213341673544308421040125, 5.24234074984832375737068861240, 6.29891348550666151603110245644, 7.382280449215646291016954671554, 8.20375293580467535302617648045, 8.79054241736173088035198338440, 9.67966465767767990303299540248, 10.31263805095956022518894116549, 12.07260830213436745480201824401, 12.933568756742411012984220347873, 13.41298569875052666674835638502, 14.205639194336228969427873413819, 15.08788500809950604971959656099, 15.59619756242645509190434157395, 16.57717491389429226786647912325, 17.29886747617671858891202478076, 17.99112182818617338883209221496, 19.115196082045804308576512366956, 19.89966422460865167024799580067, 20.79793491685642033282600176682, 21.59212384899956254845175677076, 21.96771437995367839067437566565

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