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Properties

Label 1-847-847.207-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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.905 − 0.424i)2-s + (−0.978 + 0.207i)3-s + (0.640 + 0.768i)4-s + (0.449 − 0.893i)5-s + (0.974 + 0.226i)6-s + (−0.254 − 0.967i)8-s + (0.913 − 0.406i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (−0.179 + 0.983i)16-s + (0.00951 + 0.999i)17-s + (−0.999 − 0.0190i)18-s + (0.953 − 0.299i)19-s + (0.974 − 0.226i)20-s + ⋯
L(s)  = 1  + (−0.905 − 0.424i)2-s + (−0.978 + 0.207i)3-s + (0.640 + 0.768i)4-s + (0.449 − 0.893i)5-s + (0.974 + 0.226i)6-s + (−0.254 − 0.967i)8-s + (0.913 − 0.406i)9-s + (−0.786 + 0.618i)10-s + (−0.786 − 0.618i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (−0.179 + 0.983i)16-s + (0.00951 + 0.999i)17-s + (−0.999 − 0.0190i)18-s + (0.953 − 0.299i)19-s + (0.974 − 0.226i)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(207,⋅)\chi_{847} (207, \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 0.7199716971−0.4066803346i0.7199716971 - 0.4066803346i
L(12)L(\frac12) ≈\approx 0.7199716971−0.4066803346i0.7199716971 - 0.4066803346i
L(1)L(1) ≈\approx 0.6363767816−0.1913326036i0.6363767816 - 0.1913326036i
L(1)L(1) ≈\approx 0.6363767816−0.1913326036i0.6363767816 - 0.1913326036i

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+(−0.905−0.424i)T 1 + (-0.905 - 0.424i)T
3 1+(−0.978+0.207i)T 1 + (-0.978 + 0.207i)T
5 1+(0.449−0.893i)T 1 + (0.449 - 0.893i)T
13 1+(0.696−0.717i)T 1 + (0.696 - 0.717i)T
17 1+(0.00951+0.999i)T 1 + (0.00951 + 0.999i)T
19 1+(0.953−0.299i)T 1 + (0.953 - 0.299i)T
23 1+(0.723−0.690i)T 1 + (0.723 - 0.690i)T
29 1+(0.993+0.113i)T 1 + (0.993 + 0.113i)T
31 1+(0.999−0.0380i)T 1 + (0.999 - 0.0380i)T
37 1+(−0.0665+0.997i)T 1 + (-0.0665 + 0.997i)T
41 1+(−0.921+0.389i)T 1 + (-0.921 + 0.389i)T
43 1+(0.841−0.540i)T 1 + (0.841 - 0.540i)T
47 1+(−0.999+0.0190i)T 1 + (-0.999 + 0.0190i)T
53 1+(−0.179−0.983i)T 1 + (-0.179 - 0.983i)T
59 1+(0.123+0.992i)T 1 + (0.123 + 0.992i)T
61 1+(0.820+0.572i)T 1 + (0.820 + 0.572i)T
67 1+(−0.327+0.945i)T 1 + (-0.327 + 0.945i)T
71 1+(0.198+0.980i)T 1 + (0.198 + 0.980i)T
73 1+(−0.851+0.524i)T 1 + (-0.851 + 0.524i)T
79 1+(0.988+0.151i)T 1 + (0.988 + 0.151i)T
83 1+(0.610−0.791i)T 1 + (0.610 - 0.791i)T
89 1+(−0.995−0.0950i)T 1 + (-0.995 - 0.0950i)T
97 1+(−0.998−0.0570i)T 1 + (-0.998 - 0.0570i)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

−22.47458462006538631019180231594, −21.38859732444335718328604351306, −20.74840428523875935999003386470, −19.3669073218981970828622911156, −18.85240948342509159220227181610, −18.00296627327045791607165445148, −17.726782579992833245383852579378, −16.71963191782407205498690924101, −15.98860777394384343118171711530, −15.35917363245651776105949244211, −14.10094729294623455334154446403, −13.63051319951847688572013564865, −12.10660172114480394065925118055, −11.35767976318189550773284012421, −10.81720434662167717262993376480, −9.86722717702013561644627100486, −9.28497722083751375943215887161, −7.91551207646482406564276433546, −7.07190377194313479266972991202, −6.51314908634835465543663585469, −5.71392528606277697072354451402, −4.82610425904666123025489478697, −3.1834121294660055405636721042, −1.9606082583537887920301331195, −0.96959469795578676437417033011, 0.82110696530715119649623668332, 1.439387340552523432651534321191, 2.918350703973749498505076785800, 4.12159552420718608544769169698, 5.12940150530554011241068621152, 6.09228502712370163059621900312, 6.8783770700613350173965039884, 8.1811168648284085706995411268, 8.75322991433109052417882736097, 9.92834292092710876532303514904, 10.29225321767549694192622702598, 11.30939032222343535307972796372, 12.04943223337440161498745447110, 12.824278843894131446115140360585, 13.4517284903637996291286043962, 15.13865562672611421265781960363, 15.96426972311946945959366004914, 16.50941629873128209865829765462, 17.4324027615489850324264389821, 17.7063772259317313670378454907, 18.63697139574115765874567576964, 19.5649600387194410784534206070, 20.575732863777195607672356366210, 20.97658413522406274634850732464, 21.83421201371513400561794337474

Graph of the ZZ-function along the critical line