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Properties

Label 1-847-847.222-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.0592 - 0.998i$
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  + (0.625 − 0.780i)2-s + (−0.669 − 0.743i)3-s + (−0.217 − 0.976i)4-s + (−0.272 + 0.962i)5-s + (−0.998 + 0.0570i)6-s + (−0.897 − 0.441i)8-s + (−0.104 + 0.994i)9-s + (0.580 + 0.814i)10-s + (−0.580 + 0.814i)12-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (−0.905 + 0.424i)16-s + (0.380 + 0.924i)17-s + (0.710 + 0.703i)18-s + (0.997 + 0.0760i)19-s + (0.998 + 0.0570i)20-s + ⋯
L(s)  = 1  + (0.625 − 0.780i)2-s + (−0.669 − 0.743i)3-s + (−0.217 − 0.976i)4-s + (−0.272 + 0.962i)5-s + (−0.998 + 0.0570i)6-s + (−0.897 − 0.441i)8-s + (−0.104 + 0.994i)9-s + (0.580 + 0.814i)10-s + (−0.580 + 0.814i)12-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (−0.905 + 0.424i)16-s + (0.380 + 0.924i)17-s + (0.710 + 0.703i)18-s + (0.997 + 0.0760i)19-s + (0.998 + 0.0570i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0592 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0592 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.0592 - 0.998i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.0592 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9905616975 - 1.051057361i\)
\(L(\frac12)\) \(\approx\) \(0.9905616975 - 1.051057361i\)
\(L(1)\) \(\approx\) \(0.9517821003 - 0.6047818339i\)
\(L(1)\) \(\approx\) \(0.9517821003 - 0.6047818339i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.625 - 0.780i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (-0.272 + 0.962i)T \)
13 \( 1 + (0.198 - 0.980i)T \)
17 \( 1 + (0.380 + 0.924i)T \)
19 \( 1 + (0.997 + 0.0760i)T \)
23 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.0285 + 0.999i)T \)
31 \( 1 + (-0.00951 + 0.999i)T \)
37 \( 1 + (-0.398 - 0.917i)T \)
41 \( 1 + (0.774 - 0.633i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.710 - 0.703i)T \)
53 \( 1 + (-0.905 - 0.424i)T \)
59 \( 1 + (0.935 - 0.353i)T \)
61 \( 1 + (0.988 - 0.151i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (0.797 - 0.603i)T \)
79 \( 1 + (-0.999 + 0.0380i)T \)
83 \( 1 + (0.974 + 0.226i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (-0.696 + 0.717i)T \)
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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

−22.54346304658413877661281688638, −21.573999895813080100515224931812, −20.81968020792936513430250752177, −20.50287748529628350443637761750, −19.03011830136210544907083702315, −17.990742469790812290756622557, −17.10018265065373104265009908429, −16.56824674105368508184419038729, −15.96372016266861573449168736074, −15.33751740899617497540186177888, −14.33238428386516040669218732750, −13.48202602932357444668705319944, −12.57828510937240198784986552960, −11.680068652375388478998200799498, −11.351706952997193039820491813751, −9.59957641735040017892002369328, −9.24711909503057793882242685894, −8.14397953618157783949110412044, −7.17826398253608443615241054257, −6.176908202740415011182461883625, −5.327411101996182727334266966491, −4.636474480562988387467420123146, −4.00298419229346362909570053963, −2.8765071946358117196267163953, −0.919969034162529840579324064393, 0.81410315316467920228259857230, 1.920788613091230322069041124, 3.01266440063650662006649816258, 3.72305881142177279917256189257, 5.19532239573454419614564145611, 5.71304271853166597582681513182, 6.75878942635978635291993123945, 7.45943263218655433828878942036, 8.654701450924740015505957929730, 10.07187206499240687547995894782, 10.7332746259384761257728924215, 11.22577937810044104306907966730, 12.27044274694262053055261328616, 12.71559269812587250121957396424, 13.73609546211064886958316517654, 14.396137749811103387453858354575, 15.30180286202705522346589992253, 16.13962809561710648899601983764, 17.55027624639307193097682186234, 18.00923369661235823183557617475, 18.895656250984462587996626658225, 19.38678743299615863596149124849, 20.21790241704851889203455928280, 21.28856853013387449385080808689, 22.16973233599028896596051968886

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