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Properties

Label 1-847-847.552-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.427 + 0.904i$
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  + (−0.974 + 0.226i)2-s + (−0.309 + 0.951i)3-s + (0.897 − 0.441i)4-s + (0.362 − 0.931i)5-s + (0.0855 − 0.996i)6-s + (−0.774 + 0.633i)8-s + (−0.809 − 0.587i)9-s + (−0.142 + 0.989i)10-s + (0.142 + 0.989i)12-s + (−0.466 + 0.884i)13-s + (0.774 + 0.633i)15-s + (0.610 − 0.791i)16-s + (0.198 + 0.980i)17-s + (0.921 + 0.389i)18-s + (0.993 − 0.113i)19-s + (−0.0855 − 0.996i)20-s + ⋯
L(s)  = 1  + (−0.974 + 0.226i)2-s + (−0.309 + 0.951i)3-s + (0.897 − 0.441i)4-s + (0.362 − 0.931i)5-s + (0.0855 − 0.996i)6-s + (−0.774 + 0.633i)8-s + (−0.809 − 0.587i)9-s + (−0.142 + 0.989i)10-s + (0.142 + 0.989i)12-s + (−0.466 + 0.884i)13-s + (0.774 + 0.633i)15-s + (0.610 − 0.791i)16-s + (0.198 + 0.980i)17-s + (0.921 + 0.389i)18-s + (0.993 − 0.113i)19-s + (−0.0855 − 0.996i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.427 + 0.904i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (552, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.427 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3390247254 + 0.5351383627i\)
\(L(\frac12)\) \(\approx\) \(0.3390247254 + 0.5351383627i\)
\(L(1)\) \(\approx\) \(0.5771233460 + 0.2247910429i\)
\(L(1)\) \(\approx\) \(0.5771233460 + 0.2247910429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.974 + 0.226i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.362 - 0.931i)T \)
13 \( 1 + (-0.466 + 0.884i)T \)
17 \( 1 + (0.198 + 0.980i)T \)
19 \( 1 + (0.993 - 0.113i)T \)
23 \( 1 + (-0.959 + 0.281i)T \)
29 \( 1 + (0.736 - 0.676i)T \)
31 \( 1 + (-0.696 + 0.717i)T \)
37 \( 1 + (-0.985 + 0.170i)T \)
41 \( 1 + (0.516 + 0.856i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (0.921 - 0.389i)T \)
53 \( 1 + (0.610 + 0.791i)T \)
59 \( 1 + (-0.516 + 0.856i)T \)
61 \( 1 + (0.974 + 0.226i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (-0.564 - 0.825i)T \)
79 \( 1 + (0.998 + 0.0570i)T \)
83 \( 1 + (0.941 - 0.336i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
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

−22.130465776183390973694405434832, −20.766119641133268700961556398452, −20.04503056502230867710045766506, −19.28070668951082711252457887886, −18.44831856321730036402479273066, −18.01097751816686997324577255392, −17.46956576528542260768652680158, −16.46798917086802222534685043086, −15.63503531343048912402150064249, −14.493571025221025671007435416708, −13.77853336740752432110318717635, −12.66836569366856455842282733453, −11.931006649658369941340653871806, −11.19210091420414147779375006286, −10.347148584742802347855106353548, −9.65266103415631027414330531914, −8.48639869119440550278000109286, −7.48299626248737723772482358895, −7.16901496218851094350816097128, −6.13016643010604636146621822214, −5.34220454589053918990529276643, −3.34936088492423291795687583653, −2.62082775708309972698378808221, −1.74394940914840825623557005030, −0.449075621877489685960249046364, 1.13197880857224338780568312431, 2.23926458988768397232836053792, 3.64640392287410479660183918322, 4.75636982039351357413123048786, 5.60474638678716244633140760179, 6.34937439720944085546747763154, 7.586347963372519848237012361421, 8.62779272418111197195588260214, 9.16480147067361806731583245330, 9.985326130007620305694477855440, 10.52487837798280544879215016106, 11.86480015599635179713218046358, 12.05246255449001585517950408907, 13.661297120710023262836369954, 14.53758585686747970109844245977, 15.4654484309682307704710626883, 16.22157177146589975269254350237, 16.69318132938300832518251816503, 17.451067885768740881154263572224, 18.05653907710650391477529126071, 19.33866300450442474534179661962, 19.96638399522735795317697825273, 20.70624564197176854617065201087, 21.45895873304456118795072312747, 22.029321636389523229848979162635

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