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Properties

Label 1-847-847.468-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: $\chi_{847} (468, \cdot )$
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

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

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