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Properties

Label 1-812-812.303-r1-0-0
Degree $1$
Conductor $812$
Sign $0.275 - 0.961i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ 0.275 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.779256307 - 2.093602176i\)
\(L(\frac12)\) \(\approx\) \(2.779256307 - 2.093602176i\)
\(L(1)\) \(\approx\) \(1.648683206 - 0.3769454436i\)
\(L(1)\) \(\approx\) \(1.648683206 - 0.3769454436i\)

Euler product

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

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 $Z$-function along the critical line