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Properties

Label 1-847-847.291-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.457 - 0.889i$
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  + (−0.969 + 0.244i)2-s + (0.669 − 0.743i)3-s + (0.879 − 0.475i)4-s + (0.00951 − 0.999i)5-s + (−0.466 + 0.884i)6-s + (−0.736 + 0.676i)8-s + (−0.104 − 0.994i)9-s + (0.235 + 0.971i)10-s + (0.235 − 0.971i)12-s + (0.610 − 0.791i)13-s + (−0.736 − 0.676i)15-s + (0.548 − 0.836i)16-s + (0.820 − 0.572i)17-s + (0.345 + 0.938i)18-s + (0.797 − 0.603i)19-s + (−0.466 − 0.884i)20-s + ⋯
L(s)  = 1  + (−0.969 + 0.244i)2-s + (0.669 − 0.743i)3-s + (0.879 − 0.475i)4-s + (0.00951 − 0.999i)5-s + (−0.466 + 0.884i)6-s + (−0.736 + 0.676i)8-s + (−0.104 − 0.994i)9-s + (0.235 + 0.971i)10-s + (0.235 − 0.971i)12-s + (0.610 − 0.791i)13-s + (−0.736 − 0.676i)15-s + (0.548 − 0.836i)16-s + (0.820 − 0.572i)17-s + (0.345 + 0.938i)18-s + (0.797 − 0.603i)19-s + (−0.466 − 0.884i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.457 - 0.889i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.457 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6389930957 - 1.047546514i\)
\(L(\frac12)\) \(\approx\) \(0.6389930957 - 1.047546514i\)
\(L(1)\) \(\approx\) \(0.8247316152 - 0.4437314238i\)
\(L(1)\) \(\approx\) \(0.8247316152 - 0.4437314238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.969 + 0.244i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.00951 - 0.999i)T \)
13 \( 1 + (0.610 - 0.791i)T \)
17 \( 1 + (0.820 - 0.572i)T \)
19 \( 1 + (0.797 - 0.603i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.516 + 0.856i)T \)
31 \( 1 + (-0.761 - 0.647i)T \)
37 \( 1 + (-0.432 - 0.901i)T \)
41 \( 1 + (0.897 - 0.441i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.345 - 0.938i)T \)
53 \( 1 + (0.548 + 0.836i)T \)
59 \( 1 + (-0.0665 - 0.997i)T \)
61 \( 1 + (0.272 + 0.962i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (-0.710 + 0.703i)T \)
79 \( 1 + (-0.948 - 0.318i)T \)
83 \( 1 + (-0.362 - 0.931i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (-0.870 - 0.491i)T \)
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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

−22.11903914770446336524897211937, −21.28158012510454988298316767174, −20.82257969984498360356085329410, −19.89390593576536229511472856122, −19.00750741663016548781886636151, −18.69535977335137850527077073506, −17.70093357302215216733378531581, −16.64536779752580576448861661845, −16.081399625837635342691454024654, −15.22188446409730494125743664127, −14.44329459968027588849483129052, −13.75000148084577049680820850318, −12.37637192382994078929140215346, −11.426362751800229752996819703423, −10.6387093668525352317448038491, −10.07871127208420840236200061697, −9.29531307265270988936482630554, −8.37144931793628179402131838879, −7.67214104333794028845178324512, −6.709946656416287737760464221221, −5.74501237006837999152940541056, −4.12329422858834410828920325075, −3.36431942105912104221580811483, −2.55066176352686663082444444514, −1.520205153533330165300865932435, 0.76445030832716994458916644375, 1.41561017754134752980517054541, 2.625633987881450390356012792139, 3.622691260474285052097960363207, 5.310219964994201088243064241798, 5.93449258067265799911946605637, 7.32342730485097432377949722138, 7.65254486471664562873140283796, 8.69870119148814423162955662447, 9.16846652804527744505324629449, 10.01333588777455218230281144001, 11.26617005505853179965454310611, 12.07606661853671221573287021076, 12.89584467578193255032920820548, 13.75954784936439819413055670255, 14.651917203595385282236029284776, 15.70269615851201939410822187215, 16.16974401154662847195841014168, 17.30016532735074752932923343712, 17.89205508728925608665803944022, 18.57639997637224309934069017558, 19.54681801239770284122324021534, 20.0804996096115589903423684147, 20.66268882912444537657869857096, 21.43385447199463410204862442594

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