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Properties

Label 1-847-847.677-r0-0-0
Degree $1$
Conductor $847$
Sign $0.999 + 0.0151i$
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  + (−0.272 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.999 + 0.0151i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.999 + 0.0151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.281470453 + 0.009679788428i\)
\(L(\frac12)\) \(\approx\) \(1.281470453 + 0.009679788428i\)
\(L(1)\) \(\approx\) \(1.000647573 + 0.2543476693i\)
\(L(1)\) \(\approx\) \(1.000647573 + 0.2543476693i\)

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 \( 1 + (0.870 + 0.491i)T \)
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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

−21.73276326188559980350596729414, −21.3697226896787384807245349617, −20.06824740737868478353717893346, −19.86315002641378179201685199077, −19.16896231105586844739352157253, −18.26634966157291291492143792624, −17.88180370602543000077220638120, −16.40829712128706792622431288242, −15.62608804825923751551198959275, −14.66514900998012224968352071583, −13.92368545239024537587856983800, −13.14697403136757984950281078765, −12.34253646576450775612179482094, −11.351044208063324640291996752790, −10.83451950804992639784667362542, −9.66587445407336242710284992669, −8.926949086205094610544739648043, −8.2129811893415275152238459166, −7.391008138590700964281961109577, −6.476644577528690003485970572457, −4.610940445153422116069394721542, −3.967408437184470279047711766, −3.13396366426880318109990998962, −2.28710257411014593008940607188, −1.21253930766514708724360846452, 0.63989759716959922222026318664, 2.09854033363075700098370325049, 3.6634740292070939464195573191, 4.15999410258952423346641180216, 5.19532340181613918261789778759, 6.292542410131262949968541079958, 7.45061327556096719156092015636, 8.066881346847103468703451619441, 8.57475946916984020622308483160, 9.50771196385615666221914667596, 10.28466413455386713571284887909, 11.4387906055859880195311759520, 12.75999498844180921728897387278, 13.32244629429130678895355772498, 14.24126138569267109018693659775, 15.07695414630170125910181346193, 15.67141527588758642794815967232, 16.217816305365017175737955843152, 17.106785112617227537514673601704, 18.33296237396175081055581794437, 18.73729755400421341963655927943, 19.843208979250234861831457363013, 20.22894990957174990742178737325, 21.1560461985861920011288387360, 22.43343347801559996507903580002

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