↑

Properties

Label 1-633-633.287-r1-0-0
Degree $1$
Conductor $633$
Sign $0.269 - 0.962i$
Analytic cond. $68.0252$
Root an. cond. $68.0252$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0448 + 0.998i)2-s + (−0.995 + 0.0896i)4-s + (−0.936 − 0.351i)5-s + (−0.0448 + 0.998i)7-s + (−0.134 − 0.990i)8-s + (0.309 − 0.951i)10-s + (0.550 − 0.834i)11-s + (0.983 − 0.178i)13-s − 14-s + (0.983 − 0.178i)16-s + (−0.473 + 0.880i)17-s + (−0.809 + 0.587i)19-s + (0.963 + 0.266i)20-s + (0.858 + 0.512i)22-s + (0.809 + 0.587i)23-s + ⋯
L(s)  = 1  + (0.0448 + 0.998i)2-s + (−0.995 + 0.0896i)4-s + (−0.936 − 0.351i)5-s + (−0.0448 + 0.998i)7-s + (−0.134 − 0.990i)8-s + (0.309 − 0.951i)10-s + (0.550 − 0.834i)11-s + (0.983 − 0.178i)13-s − 14-s + (0.983 − 0.178i)16-s + (−0.473 + 0.880i)17-s + (−0.809 + 0.587i)19-s + (0.963 + 0.266i)20-s + (0.858 + 0.512i)22-s + (0.809 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.269 - 0.962i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ 0.269 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1023089453 - 0.07757047801i\)
\(L(\frac12)\) \(\approx\) \(0.1023089453 - 0.07757047801i\)
\(L(1)\) \(\approx\) \(0.6336969280 + 0.3753077297i\)
\(L(1)\) \(\approx\) \(0.6336969280 + 0.3753077297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.0448 + 0.998i)T \)
5 \( 1 + (-0.936 - 0.351i)T \)
7 \( 1 + (-0.0448 + 0.998i)T \)
11 \( 1 + (0.550 - 0.834i)T \)
13 \( 1 + (0.983 - 0.178i)T \)
17 \( 1 + (-0.473 + 0.880i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.983 + 0.178i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (-0.550 - 0.834i)T \)
41 \( 1 + (0.691 + 0.722i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.753 - 0.657i)T \)
53 \( 1 + (0.995 + 0.0896i)T \)
59 \( 1 + (0.691 + 0.722i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (0.134 - 0.990i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.963 + 0.266i)T \)
97 \( 1 + (0.858 + 0.512i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.68686338711184590414492762626, −22.376206873866140237241614002772, −20.926508841546750421544840678066, −20.42237747334168905197089199746, −19.73017041092259555092511587907, −18.97141860509668026693356411000, −18.181735652920138525259295468680, −17.263838474165846339096334306049, −16.39154519749209686742852052455, −15.18553720081506525331781834980, −14.48718693591144950128313700598, −13.45764650556568453760097204580, −12.81700600996054280768958694043, −11.69068740390761745917639545442, −11.11435119918652304597468363767, −10.4413817731609925471893788233, −9.328877541771386635377685493693, −8.513922530349839843112775037966, −7.35697671640544691538333804350, −6.61603726469979580211460421706, −4.92094118406891431876633061954, −4.11836810979785919975462350248, −3.52416306015935623771569572194, −2.26433765585022455000653466982, −0.985835336678919002857652028590, 0.03799942793171787212307739565, 1.45952525804147844749022439535, 3.41418737459407400104656028371, 3.971534429962610489965456219768, 5.26428600219375268404890026865, 5.98711764727608466152393935994, 6.895022344471520227716646533621, 8.120795382806472312815429373982, 8.63199281237164166774949511349, 9.22776618570736905619974964706, 10.78711408907641144303671440706, 11.63346347021895329604794110433, 12.74728344284071164377645799121, 13.214675881968110818689610166929, 14.64554889602504741057546291911, 15.040870519191478203504869329139, 16.00055812469691773217356960640, 16.47652391366711491644062072131, 17.44898749876772120865539660919, 18.52022535230344805051558605600, 19.03486538382892969316643983051, 19.86187776935796347202497543756, 21.274506511711311778874942265371, 21.77434175117299965029600112473, 22.91076021001935068134302115468

Graph of the $Z$-function along the critical line