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Properties

Label 1-847-847.734-r0-0-0
Degree $1$
Conductor $847$
Sign $0.869 - 0.493i$
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  + (0.985 + 0.170i)2-s + (0.809 − 0.587i)3-s + (0.941 + 0.336i)4-s + (−0.993 − 0.113i)5-s + (0.897 − 0.441i)6-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (−0.959 − 0.281i)10-s + (0.959 − 0.281i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 + 0.491i)15-s + (0.774 + 0.633i)16-s + (0.516 − 0.856i)17-s + (0.466 − 0.884i)18-s + (0.0855 − 0.996i)19-s + (−0.897 − 0.441i)20-s + ⋯
L(s)  = 1  + (0.985 + 0.170i)2-s + (0.809 − 0.587i)3-s + (0.941 + 0.336i)4-s + (−0.993 − 0.113i)5-s + (0.897 − 0.441i)6-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (−0.959 − 0.281i)10-s + (0.959 − 0.281i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 + 0.491i)15-s + (0.774 + 0.633i)16-s + (0.516 − 0.856i)17-s + (0.466 − 0.884i)18-s + (0.0855 − 0.996i)19-s + (−0.897 − 0.441i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.869 - 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.186801476 - 0.8420876220i\)
\(L(\frac12)\) \(\approx\) \(3.186801476 - 0.8420876220i\)
\(L(1)\) \(\approx\) \(2.198939592 - 0.2866493501i\)
\(L(1)\) \(\approx\) \(2.198939592 - 0.2866493501i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.985 + 0.170i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.993 - 0.113i)T \)
13 \( 1 + (-0.0285 + 0.999i)T \)
17 \( 1 + (0.516 - 0.856i)T \)
19 \( 1 + (0.0855 - 0.996i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (0.564 - 0.825i)T \)
37 \( 1 + (0.610 + 0.791i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.466 + 0.884i)T \)
53 \( 1 + (0.774 - 0.633i)T \)
59 \( 1 + (-0.696 + 0.717i)T \)
61 \( 1 + (-0.985 + 0.170i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (-0.998 - 0.0570i)T \)
79 \( 1 + (0.736 + 0.676i)T \)
83 \( 1 + (-0.254 + 0.967i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (-0.993 + 0.113i)T \)
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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

−22.14560218122138300362977383176, −21.32335378266224271939894629441, −20.652487928675083692398842216349, −19.922189099470386110938750439703, −19.35508075257623322529280636016, −18.61703273754828622250575310919, −17.024951419881691042432403571292, −16.17695441360601348486222875064, −15.4886716609320737234127856465, −14.8561642822005596183168146347, −14.36435309832462400408197591459, −13.21030255562151787099085706307, −12.60332815898077398769801001075, −11.65177061143220578747869566736, −10.67196133148764324123309901196, −10.20470592939896721909718265073, −8.89666722258553612452921253566, −7.79518983135166236463670274348, −7.44205959058741127254922325670, −5.970756646512349341901270270837, −5.076866254172955674547744366418, −4.06402322169608709123879451225, −3.48321363333800301935579621483, −2.75026985610428014079583187798, −1.41469940563037970684031450401, 1.116446607620040533216348794524, 2.468854861810855580571817856961, 3.16566966637210635597804919957, 4.167039833891568876788605158273, 4.853440243153820393687562344176, 6.259940136412227692830189375045, 7.149388966743111661911565075937, 7.57931444827326012625120722335, 8.61217321459750070701582815063, 9.47835398658544782913969554006, 11.01884594623124858564723548327, 11.71495977214762278460456571798, 12.36434561235620672798485085248, 13.275509917537885067164040375733, 13.87204193400478330341311922193, 14.87541138415888748601114742337, 15.22925561807787511822937126267, 16.26753862899680684284585389410, 16.925382286551661291164565163708, 18.33883814257977946431031289389, 19.08817427852665390601705784331, 19.753342172915722970359026882836, 20.551761279632236850249896132230, 21.05824070936861168678152728043, 22.1708367044851052881454597535

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