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Properties

Label 1-847-847.600-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: $\chi_{847} (600, \cdot )$
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

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

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