↑

Properties

Label 1-60e2-3600.203-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.340 + 0.940i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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)7-s + (0.406 + 0.913i)11-s + (0.913 + 0.406i)13-s + (0.951 + 0.309i)17-s + (−0.951 − 0.309i)19-s + (0.994 + 0.104i)23-s + (−0.207 + 0.978i)29-s + (0.978 − 0.207i)31-s + (−0.809 + 0.587i)37-s + (0.913 + 0.406i)41-s + (−0.5 + 0.866i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.406 − 0.913i)59-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.406 + 0.913i)11-s + (0.913 + 0.406i)13-s + (0.951 + 0.309i)17-s + (−0.951 − 0.309i)19-s + (0.994 + 0.104i)23-s + (−0.207 + 0.978i)29-s + (0.978 − 0.207i)31-s + (−0.809 + 0.587i)37-s + (0.913 + 0.406i)41-s + (−0.5 + 0.866i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.309 + 0.951i)53-s + (0.406 − 0.913i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.340 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.723496782 + 2.456842509i\)
\(L(\frac12)\) \(\approx\) \(1.723496782 + 2.456842509i\)
\(L(1)\) \(\approx\) \(1.284806439 + 0.3790826802i\)
\(L(1)\) \(\approx\) \(1.284806439 + 0.3790826802i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.406 + 0.913i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.994 + 0.104i)T \)
29 \( 1 + (-0.207 + 0.978i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.406 - 0.913i)T \)
61 \( 1 + (-0.406 - 0.913i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.207 + 0.978i)T \)
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

−18.3149022159105679291523324129, −17.56899447102428285002870573106, −16.92176524920526800596059785706, −16.41869236332558873116576631484, −15.5170702087644929753528808461, −14.81438116574977714698549583153, −14.15640877306694866970801900318, −13.550087136219802456209792932821, −12.918966204011260364092176316005, −11.81686508166730637371250859779, −11.46849825208910261874835534565, −10.498653279689968520659686220659, −10.25194668739586858328700594312, −8.84546876090056037543010158743, −8.591367604391364002180882707715, −7.73821480252490745295062406223, −7.025738440129802991967670413221, −6.05498581761723664031869518637, −5.5200691737495492795463574783, −4.55996129616594822280677592215, −3.80107109812956119588741478900, −3.13131812265376753917786206370, −2.01031495925160856580322262850, −1.09472418169904617297992681258, −0.487342274854625617360595713400, 1.15799462117133983962077097069, 1.59581042510781856052373974712, 2.60447580458978674142711798443, 3.50809008339781980397052700293, 4.49941026672649522390627766962, 4.92394357381350015673305605674, 5.96158517960821082369052826951, 6.58292857164720622566643403573, 7.46959723353399375461550763076, 8.212615387324591584157111997640, 8.8806239667452249640035337819, 9.50712367070522297194460313951, 10.4862774707776813189332820560, 11.08872896395125870864502974624, 11.79211755166457207791245521938, 12.47141583888676869911861867533, 13.11722729336031620919928520066, 14.06013360459510120162333458167, 14.697262136009299717169719394409, 15.144462337489899311593985460119, 15.930764162095850715019000291834, 16.802317296837611819877089510744, 17.40682876304868436747074925736, 17.96851893707853073091194378188, 18.82375085771114707929789258165

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