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Properties

Label 1-847-847.240-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.410 - 0.911i$
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  + (0.449 + 0.893i)2-s + (0.669 + 0.743i)3-s + (−0.595 + 0.803i)4-s + (−0.532 + 0.846i)5-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.104 + 0.994i)9-s + (−0.995 − 0.0950i)10-s + (−0.995 + 0.0950i)12-s + (−0.870 + 0.491i)13-s + (−0.985 + 0.170i)15-s + (−0.290 − 0.956i)16-s + (−0.179 + 0.983i)17-s + (−0.935 + 0.353i)18-s + (0.879 − 0.475i)19-s + (−0.362 − 0.931i)20-s + ⋯
L(s)  = 1  + (0.449 + 0.893i)2-s + (0.669 + 0.743i)3-s + (−0.595 + 0.803i)4-s + (−0.532 + 0.846i)5-s + (−0.362 + 0.931i)6-s + (−0.985 − 0.170i)8-s + (−0.104 + 0.994i)9-s + (−0.995 − 0.0950i)10-s + (−0.995 + 0.0950i)12-s + (−0.870 + 0.491i)13-s + (−0.985 + 0.170i)15-s + (−0.290 − 0.956i)16-s + (−0.179 + 0.983i)17-s + (−0.935 + 0.353i)18-s + (0.879 − 0.475i)19-s + (−0.362 − 0.931i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.410 - 0.911i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (240, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.410 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7170994699 + 1.109244481i\)
\(L(\frac12)\) \(\approx\) \(-0.7170994699 + 1.109244481i\)
\(L(1)\) \(\approx\) \(0.5365598771 + 1.063381518i\)
\(L(1)\) \(\approx\) \(0.5365598771 + 1.063381518i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.449 + 0.893i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (-0.532 + 0.846i)T \)
13 \( 1 + (-0.870 + 0.491i)T \)
17 \( 1 + (-0.179 + 0.983i)T \)
19 \( 1 + (0.879 - 0.475i)T \)
23 \( 1 + (-0.327 + 0.945i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.749 + 0.662i)T \)
37 \( 1 + (0.953 + 0.299i)T \)
41 \( 1 + (-0.254 - 0.967i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.935 - 0.353i)T \)
53 \( 1 + (-0.290 + 0.956i)T \)
59 \( 1 + (-0.710 - 0.703i)T \)
61 \( 1 + (0.548 + 0.836i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (0.610 - 0.791i)T \)
73 \( 1 + (0.483 + 0.875i)T \)
79 \( 1 + (-0.969 - 0.244i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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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

−21.29795279861770673500783630530, −20.497704331390148026406320914895, −20.04723284107878160541430751109, −19.518784730641538530353700712506, −18.48710061835238385556540368515, −18.02314231867460363300803721138, −16.81106858244169241484195945425, −15.75412858756344411285784585936, −14.782019710868544331016527935015, −14.18640143510550755525823301187, −13.17156886745037417389358125126, −12.697112044521055650670313643195, −11.9318844778004900512908419130, −11.31118615279385267883815603645, −9.81989530816855150787402858278, −9.36742990429678020727228450433, −8.28449005260342857294780356337, −7.628090780023394880205312257486, −6.39553902233862126469535770927, −5.22544736151328252084561076823, −4.457503382918045595223628835930, −3.34715019797178535679060824032, −2.58846121594831290936700118406, −1.46153032406920544365492271674, −0.46588277584237612421492184379, 2.286377087935252589967882726678, 3.25941063410189735780009294517, 3.97070618188375961812437631721, 4.79631417728134329366092999127, 5.83497110230097208164463456030, 6.96880760235537619120946756101, 7.637916348394622170346990550591, 8.39674037921400995692748041832, 9.416618686460750654311381637489, 10.14118674146066772464824406779, 11.31662228359449064754942965843, 12.09333315165767657663726762507, 13.3545853684431894632801527867, 14.02530483783988823882942299309, 14.73716672527006718005374620827, 15.38510060083098982882443951236, 15.8699085304767983791952747627, 16.90056792846918599926664399299, 17.62179266438095694206963945189, 18.75926737064223852302623912095, 19.44514463237635752860399930806, 20.294733616778674197708966693705, 21.47427869611176782192453677603, 21.9027842710275836711188613038, 22.52418366415329942564037883103

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