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Properties

Label 1-847-847.24-r0-0-0
Degree $1$
Conductor $847$
Sign $0.0756 + 0.997i$
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  + (0.532 + 0.846i)2-s + (0.978 − 0.207i)3-s + (−0.432 + 0.901i)4-s + (0.948 + 0.318i)5-s + (0.696 + 0.717i)6-s + (−0.993 + 0.113i)8-s + (0.913 − 0.406i)9-s + (0.235 + 0.971i)10-s + (−0.235 + 0.971i)12-s + (0.941 + 0.336i)13-s + (0.993 + 0.113i)15-s + (−0.625 − 0.780i)16-s + (−0.290 − 0.956i)17-s + (0.830 + 0.556i)18-s + (−0.999 + 0.0190i)19-s + (−0.696 + 0.717i)20-s + ⋯
L(s)  = 1  + (0.532 + 0.846i)2-s + (0.978 − 0.207i)3-s + (−0.432 + 0.901i)4-s + (0.948 + 0.318i)5-s + (0.696 + 0.717i)6-s + (−0.993 + 0.113i)8-s + (0.913 − 0.406i)9-s + (0.235 + 0.971i)10-s + (−0.235 + 0.971i)12-s + (0.941 + 0.336i)13-s + (0.993 + 0.113i)15-s + (−0.625 − 0.780i)16-s + (−0.290 − 0.956i)17-s + (0.830 + 0.556i)18-s + (−0.999 + 0.0190i)19-s + (−0.696 + 0.717i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0756 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0756 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0756 + 0.997i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.0756 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.314287252 + 2.145266266i\)
\(L(\frac12)\) \(\approx\) \(2.314287252 + 2.145266266i\)
\(L(1)\) \(\approx\) \(1.830001926 + 1.030636006i\)
\(L(1)\) \(\approx\) \(1.830001926 + 1.030636006i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.532 + 0.846i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (0.948 + 0.318i)T \)
13 \( 1 + (0.941 + 0.336i)T \)
17 \( 1 + (-0.290 - 0.956i)T \)
19 \( 1 + (-0.999 + 0.0190i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (-0.380 + 0.924i)T \)
37 \( 1 + (0.879 + 0.475i)T \)
41 \( 1 + (-0.985 - 0.170i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.830 - 0.556i)T \)
53 \( 1 + (-0.625 + 0.780i)T \)
59 \( 1 + (-0.640 - 0.768i)T \)
61 \( 1 + (0.999 + 0.0380i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (0.0855 - 0.996i)T \)
73 \( 1 + (0.161 - 0.986i)T \)
79 \( 1 + (-0.00951 + 0.999i)T \)
83 \( 1 + (-0.998 + 0.0570i)T \)
89 \( 1 + (-0.981 + 0.189i)T \)
97 \( 1 + (-0.198 + 0.980i)T \)
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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

−21.606590584700752417362690359226, −21.13468602641060117140737027725, −20.44789526853391422371956437307, −19.809201302277352077147406002842, −18.883047692328852980874394456365, −18.24869970405577408510916872139, −17.25880452421350315713343552054, −16.12806582433892203805971154128, −15.10200223064006535241453180690, −14.53629621887136336291619944167, −13.663329618483094616607259826, −13.03286708879112415795992191720, −12.58034936381178918506092568269, −11.098645741741036627764758067270, −10.385887544046249511661212587182, −9.729798475358768314995815994359, −8.75496526753431940357251262399, −8.2847893769790803245799812055, −6.536331155005716848059832534139, −5.86824367603994832760834519670, −4.61806493389185028083558996970, −3.9993029603703115459162009193, −2.841651977292924460680192605610, −2.10700582267827784877200291508, −1.20134912924165319083797305903, 1.55340252042248534718664793232, 2.6718518909046430511089713567, 3.46452474942734663225391509157, 4.511717894467606937134181848849, 5.55725892670424368780968080176, 6.618280418392685856249942056653, 7.02729853697581476237674797553, 8.2377609571872044127127764768, 8.91412701220224726931047515896, 9.61760202982513548072413855547, 10.76621457761955716789728012552, 12.04142743909391908963321085279, 13.01271212451015684476607553089, 13.72032763963212133693262092071, 14.01173386635597606784460439788, 14.99916004276104454842004665948, 15.64829439511660965885074177345, 16.55768520464880830641516489070, 17.546573514194024626754694563844, 18.25237995793632695619129274053, 18.86103711278397790090432640011, 20.1254515296621048385011877813, 20.91435200839675961391802898820, 21.56230675591833552217211733894, 22.13950333584978466532644304244

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