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Properties

Label 1-847-847.135-r0-0-0
Degree 11
Conductor 847847
Sign −0.987−0.160i-0.987 - 0.160i
Analytic cond. 3.933453.93345
Root an. cond. 3.933453.93345
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0665 − 0.997i)2-s + (−0.104 − 0.994i)3-s + (−0.991 + 0.132i)4-s + (0.953 + 0.299i)5-s + (−0.985 + 0.170i)6-s + (0.198 + 0.980i)8-s + (−0.978 + 0.207i)9-s + (0.235 − 0.971i)10-s + (0.235 + 0.971i)12-s + (−0.564 + 0.825i)13-s + (0.198 − 0.980i)15-s + (0.964 − 0.263i)16-s + (0.797 − 0.603i)17-s + (0.272 + 0.962i)18-s + (−0.290 − 0.956i)19-s + (−0.985 − 0.170i)20-s + ⋯
L(s)  = 1  + (−0.0665 − 0.997i)2-s + (−0.104 − 0.994i)3-s + (−0.991 + 0.132i)4-s + (0.953 + 0.299i)5-s + (−0.985 + 0.170i)6-s + (0.198 + 0.980i)8-s + (−0.978 + 0.207i)9-s + (0.235 − 0.971i)10-s + (0.235 + 0.971i)12-s + (−0.564 + 0.825i)13-s + (0.198 − 0.980i)15-s + (0.964 − 0.263i)16-s + (0.797 − 0.603i)17-s + (0.272 + 0.962i)18-s + (−0.290 − 0.956i)19-s + (−0.985 − 0.170i)20-s + ⋯

Functional equation

Λ(s)=(847s/2 ΓR(s) L(s)=((−0.987−0.160i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(847s/2 ΓR(s) L(s)=((−0.987−0.160i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 847847    =    7⋅1127 \cdot 11^{2}
Sign: −0.987−0.160i-0.987 - 0.160i
Analytic conductor: 3.933453.93345
Root analytic conductor: 3.933453.93345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ847(135,⋅)\chi_{847} (135, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 847, (0: ), −0.987−0.160i)(1,\ 847,\ (0:\ ),\ -0.987 - 0.160i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.09433433251−1.168074204i0.09433433251 - 1.168074204i
L(12)L(\frac12) ≈\approx 0.09433433251−1.168074204i0.09433433251 - 1.168074204i
L(1)L(1) ≈\approx 0.6558956118−0.7230463185i0.6558956118 - 0.7230463185i
L(1)L(1) ≈\approx 0.6558956118−0.7230463185i0.6558956118 - 0.7230463185i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
11 1 1
good2 1+(−0.0665−0.997i)T 1 + (-0.0665 - 0.997i)T
3 1+(−0.104−0.994i)T 1 + (-0.104 - 0.994i)T
5 1+(0.953+0.299i)T 1 + (0.953 + 0.299i)T
13 1+(−0.564+0.825i)T 1 + (-0.564 + 0.825i)T
17 1+(0.797−0.603i)T 1 + (0.797 - 0.603i)T
19 1+(−0.290−0.956i)T 1 + (-0.290 - 0.956i)T
23 1+(0.0475−0.998i)T 1 + (0.0475 - 0.998i)T
29 1+(0.0855+0.996i)T 1 + (0.0855 + 0.996i)T
31 1+(−0.851−0.524i)T 1 + (-0.851 - 0.524i)T
37 1+(−0.179−0.983i)T 1 + (-0.179 - 0.983i)T
41 1+(−0.466−0.884i)T 1 + (-0.466 - 0.884i)T
43 1+(0.415−0.909i)T 1 + (0.415 - 0.909i)T
47 1+(0.272−0.962i)T 1 + (0.272 - 0.962i)T
53 1+(0.964+0.263i)T 1 + (0.964 + 0.263i)T
59 1+(−0.532−0.846i)T 1 + (-0.532 - 0.846i)T
61 1+(−0.830−0.556i)T 1 + (-0.830 - 0.556i)T
67 1+(−0.786−0.618i)T 1 + (-0.786 - 0.618i)T
71 1+(0.516−0.856i)T 1 + (0.516 - 0.856i)T
73 1+(0.988+0.151i)T 1 + (0.988 + 0.151i)T
79 1+(−0.595−0.803i)T 1 + (-0.595 - 0.803i)T
83 1+(0.774+0.633i)T 1 + (0.774 + 0.633i)T
89 1+(0.981+0.189i)T 1 + (0.981 + 0.189i)T
97 1+(−0.736−0.676i)T 1 + (-0.736 - 0.676i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−22.583204590592315604685203676035, −21.62939469601030354067734477273, −21.26868418069694639167787363758, −20.21767062860426974746097510574, −19.2380083810820274770140477147, −18.11850038564353388043011728420, −17.3851825664412521057467251648, −16.85342840595065530651661660110, −16.21012606496967330441279949619, −15.17186081517965203034906779378, −14.68697954971784757044409268087, −13.858526017866827605522670485633, −12.97700624134185534945839400729, −12.05979941865864796014017529860, −10.56591770743444808742406383607, −9.97308997370384758592304271413, −9.40344525226186500567932635414, −8.41844725614018821655112487412, −7.66997281116707744182355481161, −6.24200064616522909525285650191, −5.673844295318320006402989782097, −5.01594118440145088693191846074, −4.01248457283735095613197836962, −2.96337792975192107913685585206, −1.294153804963531686984862638282, 0.58364736251418421544149147888, 1.892787751310534355250441594165, 2.35527649486907041445913016765, 3.38959871787065130568270516312, 4.85853297534358733869962550732, 5.59868582499626356084096332255, 6.74920186971080341165625147393, 7.48590399276768510848923349798, 8.83475168956555821972318222699, 9.275890608563308912319443339986, 10.4451705311890263333531253291, 11.08084363872407409073102769701, 12.1307410049745291914374182884, 12.61494043299252954234477212334, 13.61631229210950690367582503426, 14.071630993784185486046251753924, 14.80796562538434459880911212667, 16.67129537616724818461475970842, 17.10784814231719475732423177655, 18.11806439130499038742459929766, 18.492645596775941774074342235319, 19.25751756469223436960575982183, 20.089872871994184093060262201897, 20.87943308682482323676395378661, 21.7906798374168932068203234084

Graph of the ZZ-function along the critical line