↑

Properties

Label 1-273-273.206-r1-0-0
Degree $1$
Conductor $273$
Sign $-0.635 + 0.771i$
Analytic cond. $29.3379$
Root an. cond. $29.3379$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + i·8-s + 10-s + i·11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − i·19-s + (−0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s + i·8-s + 10-s + i·11-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s − i·19-s + (−0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + (0.5 + 0.866i)25-s + (0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.635 + 0.771i$
Analytic conductor: \(29.3379\)
Root analytic conductor: \(29.3379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (1:\ ),\ -0.635 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1755923345 + 0.3720957969i\)
\(L(\frac12)\) \(\approx\) \(0.1755923345 + 0.3720957969i\)
\(L(1)\) \(\approx\) \(0.5491671437 + 0.09164948140i\)
\(L(1)\) \(\approx\) \(0.5491671437 + 0.09164948140i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + iT \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 - T \)
67 \( 1 + iT \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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

−25.50709755583315564519246698311, −24.266985418880752998636415806080, −23.41532607427431532817126431044, −22.17964148073259608128230199115, −21.47704056583086797256434761774, −20.38225226774113548154030816799, −19.452728576855320847038733122631, −18.881604248087249425201550467227, −18.06114644112452188383753504906, −16.83197279864737930064563134487, −16.13444936231000275543210303376, −15.1395570180640192648375164779, −13.952275659814271270921323161525, −12.572707668105839294675624454398, −11.783588879230838245633486063148, −10.86584772993408226243543672939, −10.137103179112275603060229871236, −8.74855276523607011841961149513, −8.01180635700030419800888529254, −7.08320245831355099265361549893, −5.81643412716745132346495274647, −3.84307290189929907294435497462, −3.26759753143802184732841803646, −1.69829580068906940489014926876, −0.20392588672883172498510746418, 1.03854293850117288411832846801, 2.57851948208465679647792466783, 4.36263445339495036660607172011, 5.316588658657636548948198260902, 6.82677304607720362732649249328, 7.5390927415859249032236711958, 8.55689716366923788755106353144, 9.44264942600462180903315651958, 10.49507954968946519922840646985, 11.62963076573297949762637812705, 12.4303896095705743848817819185, 13.90813949463024141328269284363, 15.05057918677754598403106252608, 15.72720156567823694679938421390, 16.56019670271024005728267693548, 17.5079418271135186290500298591, 18.413365399115188842358077208740, 19.37785098424248234560536654710, 20.17508064762402258186203428791, 20.81419899592119688113716672965, 22.49518117694723732155949249309, 23.33997476014876779356377723969, 24.11584567273394505449378296849, 24.96952378484457251073622229001, 25.84470490394137907701644038622

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