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Properties

Label 1-847-847.101-r0-0-0
Degree 11
Conductor 847847
Sign 0.193−0.981i0.193 - 0.981i
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  + (−0.988 − 0.151i)2-s + (0.978 − 0.207i)3-s + (0.953 + 0.299i)4-s + (0.969 − 0.244i)5-s + (−0.998 + 0.0570i)6-s + (−0.897 − 0.441i)8-s + (0.913 − 0.406i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (0.820 + 0.572i)16-s + (−0.991 − 0.132i)17-s + (−0.964 + 0.263i)18-s + (−0.432 − 0.901i)19-s + (0.998 + 0.0570i)20-s + ⋯
L(s)  = 1  + (−0.988 − 0.151i)2-s + (0.978 − 0.207i)3-s + (0.953 + 0.299i)4-s + (0.969 − 0.244i)5-s + (−0.998 + 0.0570i)6-s + (−0.897 − 0.441i)8-s + (0.913 − 0.406i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (0.820 + 0.572i)16-s + (−0.991 − 0.132i)17-s + (−0.964 + 0.263i)18-s + (−0.432 − 0.901i)19-s + (0.998 + 0.0570i)20-s + ⋯

Functional equation

Λ(s)=(847s/2 ΓR(s) L(s)=((0.193−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(847s/2 ΓR(s) L(s)=((0.193−0.981i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 847847    =    7⋅1127 \cdot 11^{2}
Sign: 0.193−0.981i0.193 - 0.981i
Analytic conductor: 3.933453.93345
Root analytic conductor: 3.933453.93345
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ847(101,⋅)\chi_{847} (101, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 847, (0: ), 0.193−0.981i)(1,\ 847,\ (0:\ ),\ 0.193 - 0.981i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.190828875−0.9793761721i1.190828875 - 0.9793761721i
L(12)L(\frac12) ≈\approx 1.190828875−0.9793761721i1.190828875 - 0.9793761721i
L(1)L(1) ≈\approx 1.067407335−0.3647650495i1.067407335 - 0.3647650495i
L(1)L(1) ≈\approx 1.067407335−0.3647650495i1.067407335 - 0.3647650495i

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+(−0.988−0.151i)T 1 + (-0.988 - 0.151i)T
3 1+(0.978−0.207i)T 1 + (0.978 - 0.207i)T
5 1+(0.969−0.244i)T 1 + (0.969 - 0.244i)T
13 1+(0.198−0.980i)T 1 + (0.198 - 0.980i)T
17 1+(−0.991−0.132i)T 1 + (-0.991 - 0.132i)T
19 1+(−0.432−0.901i)T 1 + (-0.432 - 0.901i)T
23 1+(−0.327−0.945i)T 1 + (-0.327 - 0.945i)T
29 1+(0.0285+0.999i)T 1 + (0.0285 + 0.999i)T
31 1+(−0.861−0.508i)T 1 + (-0.861 - 0.508i)T
37 1+(−0.595+0.803i)T 1 + (-0.595 + 0.803i)T
41 1+(0.774−0.633i)T 1 + (0.774 - 0.633i)T
43 1+(0.142−0.989i)T 1 + (0.142 - 0.989i)T
47 1+(−0.964−0.263i)T 1 + (-0.964 - 0.263i)T
53 1+(0.820−0.572i)T 1 + (0.820 - 0.572i)T
59 1+(−0.161+0.986i)T 1 + (-0.161 + 0.986i)T
61 1+(−0.625−0.780i)T 1 + (-0.625 - 0.780i)T
67 1+(0.0475−0.998i)T 1 + (0.0475 - 0.998i)T
71 1+(0.941−0.336i)T 1 + (0.941 - 0.336i)T
73 1+(0.123+0.992i)T 1 + (0.123 + 0.992i)T
79 1+(0.532+0.846i)T 1 + (0.532 + 0.846i)T
83 1+(0.974+0.226i)T 1 + (0.974 + 0.226i)T
89 1+(−0.235+0.971i)T 1 + (-0.235 + 0.971i)T
97 1+(−0.696+0.717i)T 1 + (-0.696 + 0.717i)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

−21.84181629859097714395810169211, −21.26120769163710827666871908064, −20.7076873428444388362492912383, −19.66728643350115596758670704533, −19.184093036649349951175710845281, −18.2570413871700900217996949491, −17.68887103583160677137494332320, −16.65541879907047796367100250179, −15.99303793538875455819661023546, −15.03077690523525046893682784434, −14.350141773341974479459111565870, −13.61140441213567484868945189670, −12.630172560079943569438917620838, −11.33408423722162353333619775315, −10.547111040275757499516471560956, −9.66525489225789555034287231283, −9.20663804682482488347556483940, −8.39305281798626206268819617613, −7.439178462226565367289450714091, −6.58353823864500320366910212504, −5.755363310792756418073046416610, −4.33068579240556610238737283262, −3.147494807575765386609164910921, −2.0365129864829458897700431919, −1.64394854759553557129052709504, 0.814357443288016097728088666390, 2.032438522941969613570390218686, 2.54907883977394656503226506557, 3.65472375622603460947394771023, 5.077555606355147927863254514218, 6.37375493727098837197956943437, 7.00012667834295810170867857278, 8.110166152168899435840269416945, 8.81918693221625210790674473597, 9.33723272246257654099756145225, 10.331873573791707056200085205667, 10.894519619638446702287546679838, 12.3573937720098116511239857682, 12.97628783743796031345010921959, 13.75495794123595043154388942299, 14.84391498220029340462774136714, 15.50084060604929620500846829939, 16.43295612871967923877141867481, 17.39419234255681945613221524754, 18.07965462822082984575869449921, 18.57244033754836414771550540786, 19.759036951341801553381044706246, 20.154469218478841077620720727185, 20.85800149939534184702082225537, 21.6271691255549862974833586278

Graph of the ZZ-function along the critical line