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Properties

Label 1-847-847.137-r0-0-0
Degree 11
Conductor 847847
Sign 0.516−0.856i0.516 - 0.856i
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  + (−0.683 + 0.730i)2-s + (0.669 − 0.743i)3-s + (−0.0665 − 0.997i)4-s + (0.988 + 0.151i)5-s + (0.0855 + 0.996i)6-s + (0.774 + 0.633i)8-s + (−0.104 − 0.994i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (−0.991 + 0.132i)16-s + (−0.948 + 0.318i)17-s + (0.797 + 0.603i)18-s + (−0.595 + 0.803i)19-s + (0.0855 − 0.996i)20-s + ⋯
L(s)  = 1  + (−0.683 + 0.730i)2-s + (0.669 − 0.743i)3-s + (−0.0665 − 0.997i)4-s + (0.988 + 0.151i)5-s + (0.0855 + 0.996i)6-s + (0.774 + 0.633i)8-s + (−0.104 − 0.994i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (−0.991 + 0.132i)16-s + (−0.948 + 0.318i)17-s + (0.797 + 0.603i)18-s + (−0.595 + 0.803i)19-s + (0.0855 − 0.996i)20-s + ⋯

Functional equation

Λ(s)=(847s/2 ΓR(s) L(s)=((0.516−0.856i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(847s/2 ΓR(s) L(s)=((0.516−0.856i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 847847    =    7⋅1127 \cdot 11^{2}
Sign: 0.516−0.856i0.516 - 0.856i
Analytic conductor: 3.933453.93345
Root analytic conductor: 3.933453.93345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ847(137,⋅)\chi_{847} (137, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 847, (0: ), 0.516−0.856i)(1,\ 847,\ (0:\ ),\ 0.516 - 0.856i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.182074432−0.6677018385i1.182074432 - 0.6677018385i
L(12)L(\frac12) ≈\approx 1.182074432−0.6677018385i1.182074432 - 0.6677018385i
L(1)L(1) ≈\approx 1.038752476−0.1282409171i1.038752476 - 0.1282409171i
L(1)L(1) ≈\approx 1.038752476−0.1282409171i1.038752476 - 0.1282409171i

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+(−0.683+0.730i)T 1 + (-0.683 + 0.730i)T
3 1+(0.669−0.743i)T 1 + (0.669 - 0.743i)T
5 1+(0.988+0.151i)T 1 + (0.988 + 0.151i)T
13 1+(−0.466−0.884i)T 1 + (-0.466 - 0.884i)T
17 1+(−0.948+0.318i)T 1 + (-0.948 + 0.318i)T
19 1+(−0.595+0.803i)T 1 + (-0.595 + 0.803i)T
23 1+(0.723−0.690i)T 1 + (0.723 - 0.690i)T
29 1+(−0.736−0.676i)T 1 + (-0.736 - 0.676i)T
31 1+(0.272−0.962i)T 1 + (0.272 - 0.962i)T
37 1+(0.640−0.768i)T 1 + (0.640 - 0.768i)T
41 1+(0.516−0.856i)T 1 + (0.516 - 0.856i)T
43 1+(0.841−0.540i)T 1 + (0.841 - 0.540i)T
47 1+(0.797−0.603i)T 1 + (0.797 - 0.603i)T
53 1+(−0.991−0.132i)T 1 + (-0.991 - 0.132i)T
59 1+(0.483−0.875i)T 1 + (0.483 - 0.875i)T
61 1+(−0.290+0.956i)T 1 + (-0.290 + 0.956i)T
67 1+(−0.327+0.945i)T 1 + (-0.327 + 0.945i)T
71 1+(−0.870+0.491i)T 1 + (-0.870 + 0.491i)T
73 1+(0.997+0.0760i)T 1 + (0.997 + 0.0760i)T
79 1+(0.449−0.893i)T 1 + (0.449 - 0.893i)T
83 1+(0.941+0.336i)T 1 + (0.941 + 0.336i)T
89 1+(−0.995−0.0950i)T 1 + (-0.995 - 0.0950i)T
97 1+(−0.362+0.931i)T 1 + (-0.362 + 0.931i)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

−21.85215039497246089525655639250, −21.351363819603573019478202383945, −20.7042485363184483850700519376, −19.83925515327542910956139467092, −19.29819517052097552031184756633, −18.30096652012942372614088077951, −17.43553089613525511787189734998, −16.80282363613506858852369227665, −16.01619260514573552679065896831, −15.000549804268979524533549835358, −13.97594090207049092584599337924, −13.361126443521248719286252752699, −12.552534105479863487467442423512, −11.19515343624099967569144900765, −10.779301111777640917745329862235, −9.60710930291099303888291772709, −9.28821758463427364593640915173, −8.638776367091651269315344389778, −7.4621197565496955947619953254, −6.51392421950951670079793210021, −4.94638489589876355673876718142, −4.370901592507739195017242821764, −3.04889859069782836409190985805, −2.37799381191003378836886730586, −1.45655759029389456014163683302, 0.69657545709015400867443705266, 1.9932421312229758631007448238, 2.53127476445803019503982945599, 4.16549958716592629325272729915, 5.5766982884543843512967803008, 6.16790305286732220401223083892, 7.044650321106447829626780380540, 7.83384435538471429350374429227, 8.712013775336942322932200800423, 9.38231263309050898841716190280, 10.24241935885045306279264928607, 11.0638185157884591618079535348, 12.57080453266962071623009017929, 13.215569361139047321997105440381, 14.02937418141726234257053463833, 14.83081889226652492368188791047, 15.28306694098959067683571207499, 16.61843056317151507676114463729, 17.45707206656524747918508379668, 17.80170870262302028419418441805, 18.82768254920812793691851740921, 19.21386163596359549752745585152, 20.37864966348259949075584790958, 20.78586223079728633980015746817, 22.17801573775370000058596318015

Graph of the ZZ-function along the critical line