↑

Properties

Label 1-4560-4560.563-r0-0-0
Degree $1$
Conductor $4560$
Sign $-0.966 + 0.255i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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  − i·7-s − i·11-s + (−0.5 − 0.866i)13-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)29-s + 31-s + 37-s + (0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.866 + 0.5i)47-s − 49-s + (−0.5 − 0.866i)53-s + (0.866 + 0.5i)59-s + (0.866 − 0.5i)61-s + ⋯
L(s)  = 1  − i·7-s − i·11-s + (−0.5 − 0.866i)13-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.5i)29-s + 31-s + 37-s + (0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (−0.866 + 0.5i)47-s − 49-s + (−0.5 − 0.866i)53-s + (0.866 + 0.5i)59-s + (0.866 − 0.5i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.966 + 0.255i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ -0.966 + 0.255i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08732775745 - 0.6711808032i\)
\(L(\frac12)\) \(\approx\) \(-0.08732775745 - 0.6711808032i\)
\(L(1)\) \(\approx\) \(0.8214098461 - 0.3018908909i\)
\(L(1)\) \(\approx\) \(0.8214098461 - 0.3018908909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−18.49047821264676406877236716672, −17.94823767223179831917532221456, −17.36851238502871945370903590212, −16.51645575619264960411504380618, −15.90818348704230586437782516512, −15.060657257466612777803738495768, −14.79756898921721222536783052692, −13.95909711517266812249182707876, −12.99378073483658700894431327838, −12.588423591795295413603095558968, −11.71972116978154873539893775299, −11.39289312975388053628190155184, −10.30108990432308580751556851404, −9.591053151629501611885814755, −9.16704665445704384809576166109, −8.23775116861269053564792995226, −7.68027852577919335704223300654, −6.60255947669689006992632508702, −6.27249962840799443136689584773, −5.281107395753895971544090525596, −4.47978526973905083383885053066, −4.02742899468094098678463603521, −2.63536619610583434534699234926, −2.27793815817885411079809523730, −1.43649915184338542819893556888, 0.19729768307914814157671631350, 0.97754186223818471557646656267, 2.0983900000110339057353790671, 2.98706388170713611758550259303, 3.712724223277994100691607054354, 4.434969205991780474917660731975, 5.29674411014504940927068329196, 6.00435408996113793537197366064, 6.8287188204306063814317810323, 7.5429556530192020811787201493, 8.12719553360135486171717505297, 8.92222108379927122901603641422, 9.80101824171387731532621339196, 10.32155386315653574727338592253, 11.18055772652618422567046573826, 11.50321038794322604231302833367, 12.64469888456186175562093117457, 13.15397367044189368860498948821, 13.887856114093121353304385525284, 14.3114061972082848856963418664, 15.25890613680051511648104163831, 15.93633746307322640628623065414, 16.48539665835652729997971155130, 17.285091305247794147879174950473, 17.74077922986971118997331875885

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