Properties

Label 1-201-201.107-r1-0-0
Degree 11
Conductor 201201
Sign 0.7090.704i0.709 - 0.704i
Analytic cond. 21.600421.6004
Root an. cond. 21.600421.6004
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.841 − 0.540i)5-s + (−0.142 + 0.989i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (0.415 + 0.909i)13-s + (0.959 + 0.281i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.142 − 0.989i)19-s + (0.654 + 0.755i)20-s + (−0.654 + 0.755i)22-s + (0.654 + 0.755i)23-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.959 − 0.281i)4-s + (−0.841 − 0.540i)5-s + (−0.142 + 0.989i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)10-s + (−0.841 − 0.540i)11-s + (0.415 + 0.909i)13-s + (0.959 + 0.281i)14-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + (−0.142 − 0.989i)19-s + (0.654 + 0.755i)20-s + (−0.654 + 0.755i)22-s + (0.654 + 0.755i)23-s + ⋯

Functional equation

Λ(s)=(201s/2ΓR(s+1)L(s)=((0.7090.704i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(201s/2ΓR(s+1)L(s)=((0.7090.704i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 201201    =    3673 \cdot 67
Sign: 0.7090.704i0.709 - 0.704i
Analytic conductor: 21.600421.6004
Root analytic conductor: 21.600421.6004
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ201(107,)\chi_{201} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 201, (1: ), 0.7090.704i)(1,\ 201,\ (1:\ ),\ 0.709 - 0.704i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.1462476990.4724760902i1.146247699 - 0.4724760902i
L(12)L(\frac12) \approx 1.1462476990.4724760902i1.146247699 - 0.4724760902i
L(1)L(1) \approx 0.80820919950.3732655514i0.8082091995 - 0.3732655514i
L(1)L(1) \approx 0.80820919950.3732655514i0.8082091995 - 0.3732655514i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
67 1 1
good2 1+(0.1420.989i)T 1 + (0.142 - 0.989i)T
5 1+(0.8410.540i)T 1 + (-0.841 - 0.540i)T
7 1+(0.142+0.989i)T 1 + (-0.142 + 0.989i)T
11 1+(0.8410.540i)T 1 + (-0.841 - 0.540i)T
13 1+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
17 1+(0.9590.281i)T 1 + (0.959 - 0.281i)T
19 1+(0.1420.989i)T 1 + (-0.142 - 0.989i)T
23 1+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
29 1T 1 - T
31 1+(0.4150.909i)T 1 + (0.415 - 0.909i)T
37 1+T 1 + T
41 1+(0.9590.281i)T 1 + (0.959 - 0.281i)T
43 1+(0.959+0.281i)T 1 + (-0.959 + 0.281i)T
47 1+(0.654+0.755i)T 1 + (0.654 + 0.755i)T
53 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
59 1+(0.415+0.909i)T 1 + (-0.415 + 0.909i)T
61 1+(0.8410.540i)T 1 + (0.841 - 0.540i)T
71 1+(0.959+0.281i)T 1 + (0.959 + 0.281i)T
73 1+(0.8410.540i)T 1 + (0.841 - 0.540i)T
79 1+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
83 1+(0.8410.540i)T 1 + (-0.841 - 0.540i)T
89 1+(0.6540.755i)T 1 + (0.654 - 0.755i)T
97 1+T 1 + T
show more
show less
   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−26.65783135394762275626955221493, −25.86476904099438104965386597267, −24.96946964775565187096361668628, −23.63288123698428726024814303745, −23.152247135534811170447869919694, −22.62013512878655759130813198566, −21.11844326223632057193038128885, −20.09411275820296425902411702186, −18.83022229301939483354488957514, −18.12974292459116676287135433485, −16.912555324432205463639902099855, −16.13245872301576382978789307084, −15.110518530283715655972809978670, −14.43676863081987731301125856267, −13.1914565348247579982771239711, −12.37939429445369669136847428818, −10.7237973643220412130378440561, −9.98245058090883535650531143572, −8.222128581487104733225169195105, −7.659556199024587608930365567245, −6.71094024114766304814003515674, −5.39772863040307031805469259814, −4.117808813969726399929157892350, −3.21550359017672807966267793389, −0.62044266161014960683572930918, 0.83518459625097134655474403359, 2.459049809441171939378033500121, 3.57851500082463261216568641220, 4.826216491778506748182979619010, 5.78481496443997430672018049139, 7.72075166586118105697388151180, 8.80399116157203896722261984291, 9.51894839612342809859266740051, 11.13892269573694015038976248710, 11.640994185884680893324283400248, 12.7145029648627178961967999034, 13.49010155458249238711945575988, 14.88483886224965438993566863013, 15.81166560551020960685667124001, 16.92722293434075628475539857487, 18.46124614261334904090918431589, 18.900806721882261665817642438105, 19.82289338460632055699920443855, 21.023026303909956289424846819033, 21.46278454059425829987353007903, 22.69992535220316300313935682770, 23.61416726254760323013492137345, 24.29123844349032713793146469329, 25.780796881346326522292441935844, 26.76520049557103919897558057246

Graph of the ZZ-function along the critical line