Properties

Label 1-812-812.303-r1-0-0
Degree 11
Conductor 812812
Sign 0.2750.961i0.275 - 0.961i
Analytic cond. 87.261587.2615
Root an. cond. 87.261587.2615
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  + (0.955 + 0.294i)3-s + (0.365 − 0.930i)5-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)15-s + (0.5 − 0.866i)17-s + (0.955 − 0.294i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)27-s + (−0.988 + 0.149i)31-s + (0.955 − 0.294i)33-s + (−0.826 − 0.563i)37-s + (−0.733 − 0.680i)39-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)3-s + (0.365 − 0.930i)5-s + (0.826 + 0.563i)9-s + (0.826 − 0.563i)11-s + (−0.900 − 0.433i)13-s + (0.623 − 0.781i)15-s + (0.5 − 0.866i)17-s + (0.955 − 0.294i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (0.623 + 0.781i)27-s + (−0.988 + 0.149i)31-s + (0.955 − 0.294i)33-s + (−0.826 − 0.563i)37-s + (−0.733 − 0.680i)39-s + ⋯

Functional equation

Λ(s)=(812s/2ΓR(s+1)L(s)=((0.2750.961i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(812s/2ΓR(s+1)L(s)=((0.2750.961i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 812812    =    227292^{2} \cdot 7 \cdot 29
Sign: 0.2750.961i0.275 - 0.961i
Analytic conductor: 87.261587.2615
Root analytic conductor: 87.261587.2615
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ812(303,)\chi_{812} (303, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 812, (1: ), 0.2750.961i)(1,\ 812,\ (1:\ ),\ 0.275 - 0.961i)

Particular Values

L(12)L(\frac{1}{2}) \approx 2.7792563072.093602176i2.779256307 - 2.093602176i
L(12)L(\frac12) \approx 2.7792563072.093602176i2.779256307 - 2.093602176i
L(1)L(1) \approx 1.6486832060.3769454436i1.648683206 - 0.3769454436i
L(1)L(1) \approx 1.6486832060.3769454436i1.648683206 - 0.3769454436i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
29 1 1
good3 1+(0.955+0.294i)T 1 + (0.955 + 0.294i)T
5 1+(0.3650.930i)T 1 + (0.365 - 0.930i)T
11 1+(0.8260.563i)T 1 + (0.826 - 0.563i)T
13 1+(0.9000.433i)T 1 + (-0.900 - 0.433i)T
17 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
19 1+(0.9550.294i)T 1 + (0.955 - 0.294i)T
23 1+(0.988+0.149i)T 1 + (0.988 + 0.149i)T
31 1+(0.988+0.149i)T 1 + (-0.988 + 0.149i)T
37 1+(0.8260.563i)T 1 + (-0.826 - 0.563i)T
41 1T 1 - T
43 1+(0.6230.781i)T 1 + (0.623 - 0.781i)T
47 1+(0.0747+0.997i)T 1 + (0.0747 + 0.997i)T
53 1+(0.988+0.149i)T 1 + (-0.988 + 0.149i)T
59 1+(0.50.866i)T 1 + (0.5 - 0.866i)T
61 1+(0.7330.680i)T 1 + (0.733 - 0.680i)T
67 1+(0.0747+0.997i)T 1 + (-0.0747 + 0.997i)T
71 1+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
73 1+(0.3650.930i)T 1 + (-0.365 - 0.930i)T
79 1+(0.826+0.563i)T 1 + (0.826 + 0.563i)T
83 1+(0.2220.974i)T 1 + (0.222 - 0.974i)T
89 1+(0.365+0.930i)T 1 + (-0.365 + 0.930i)T
97 1+(0.2220.974i)T 1 + (0.222 - 0.974i)T
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   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

−22.12083950625746352617123347028, −21.40412437679448762031087933600, −20.53357814623960111668846406075, −19.65235724943164364496368618908, −19.07685642936322643519720914950, −18.3559953556651450103231002325, −17.478050355937143477770729069, −16.68144199492574638238237610784, −15.34747325585463134316198445169, −14.70022509138032504609082557771, −14.30131846235191895303422088518, −13.39486318600074559160401853499, −12.44077010145782319553011300192, −11.66252378602445312831813453881, −10.42878155669763245259013008530, −9.70056991079299247066070913459, −9.05472104924354424360171822024, −7.859391657757526256784113608298, −7.09057376059585203278588196370, −6.52596520411397194663011676671, −5.20874828892263654349751891516, −3.9107253706552293851204768382, −3.1797801143186443149047399181, −2.15053661285190689974358984509, −1.353516351611056874389201440989, 0.666671374404031631885141554143, 1.69152201370981552138503316596, 2.8701729698195846922377859174, 3.70567542902910131476718992074, 4.90239695481590260526080232811, 5.43177828437899825075864148006, 6.953793648883783708125651071520, 7.734318622810196408116854834188, 8.76445054390579228786895807953, 9.33185942423025809970228803184, 9.93944605381582931250277082893, 11.16617289736356318741327177811, 12.23333349833468103548897547902, 12.95162764258440326895391874600, 13.93474434068898051248368338560, 14.32560018937404542098970313758, 15.44141080741604803866723334336, 16.18912394021805174706166968267, 16.92305646600495801847851295374, 17.76689460024811629868874101961, 18.93786480383430120452359569892, 19.55658678593973801709994587504, 20.44302113909087127849095480960, 20.76960756157092286914388526889, 21.893373726725706233921904817938

Graph of the ZZ-function along the critical line