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Properties

Label 1-847-847.221-r0-0-0
Degree $1$
Conductor $847$
Sign $0.989 - 0.145i$
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  + (−0.327 − 0.945i)2-s + (−0.5 − 0.866i)3-s + (−0.786 + 0.618i)4-s + (0.0475 + 0.998i)5-s + (−0.654 + 0.755i)6-s + (0.841 + 0.540i)8-s + (−0.5 + 0.866i)9-s + (0.928 − 0.371i)10-s + (0.928 + 0.371i)12-s + (−0.142 − 0.989i)13-s + (0.841 − 0.540i)15-s + (0.235 − 0.971i)16-s + (−0.995 + 0.0950i)17-s + (0.981 + 0.189i)18-s + (−0.995 − 0.0950i)19-s + (−0.654 − 0.755i)20-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (−0.5 − 0.866i)3-s + (−0.786 + 0.618i)4-s + (0.0475 + 0.998i)5-s + (−0.654 + 0.755i)6-s + (0.841 + 0.540i)8-s + (−0.5 + 0.866i)9-s + (0.928 − 0.371i)10-s + (0.928 + 0.371i)12-s + (−0.142 − 0.989i)13-s + (0.841 − 0.540i)15-s + (0.235 − 0.971i)16-s + (−0.995 + 0.0950i)17-s + (0.981 + 0.189i)18-s + (−0.995 − 0.0950i)19-s + (−0.654 − 0.755i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.989 - 0.145i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.989 - 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6307393104 - 0.04599695429i\)
\(L(\frac12)\) \(\approx\) \(0.6307393104 - 0.04599695429i\)
\(L(1)\) \(\approx\) \(0.5898057579 - 0.2570590422i\)
\(L(1)\) \(\approx\) \(0.5898057579 - 0.2570590422i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.327 - 0.945i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.0475 + 0.998i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.995 + 0.0950i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
23 \( 1 + (0.235 - 0.971i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.786 - 0.618i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.981 - 0.189i)T \)
53 \( 1 + (0.235 + 0.971i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.723 + 0.690i)T \)
79 \( 1 + (0.0475 + 0.998i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−22.1818450350357194836462550079, −21.40997381732650089353366412473, −20.68986492065249348302145906081, −19.6183564830663099631936843407, −18.92997099172428713577513241398, −17.48884499865422962978056200200, −17.33790705828990163864389868192, −16.55571241003320659578592610151, −15.64026535520596425618718553978, −15.41173833383815985368131184438, −14.15018136392200733091178032497, −13.46303034351165941364145778766, −12.3515966439005762780225593741, −11.44850708049121263332706426322, −10.404098553050819747938247439015, −9.56055161295669881943017814959, −8.91070459127439681515863584372, −8.28503794604953793477824452354, −6.91570906942336997405966403402, −6.16816955526348332437366676437, −5.20062638108713896521296309614, −4.54115330314468145048473252061, −3.86631878819038486886764699377, −1.88657648424133227690416156103, −0.43846212806859348492954404158, 0.93172174988561045233392344465, 2.32230559940125609958270873724, 2.700730159557852626762795375370, 4.03736796391905070092118364872, 5.161952603319494462426808698507, 6.32651837435882519859722372167, 7.08844202917296618970429159008, 8.055090665287738670962443709483, 8.79659839652853764791573846019, 10.203283841174660138635681320744, 10.71750461480842873848910332276, 11.32658580387190605501684406031, 12.34911586418430850730167543864, 12.923518298117792681999765822, 13.761844071012070559990129617637, 14.5848887555796492417683695713, 15.68751076096367520068749420390, 17.02511657666268175963895346163, 17.531395337242700896821954328693, 18.253897137440929055613473104762, 18.83042337846896754762724486201, 19.59740217961689601329081480081, 20.23152312121883474431157860889, 21.43661489672373816591661634098, 22.13266644919904721546801316666

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