↑

Properties

Label 2-84-21.11-c0-0-0
Degree 22
Conductor 8484
Sign 0.895−0.444i0.895 - 0.444i
Analytic cond. 0.04192140.0419214
Root an. cond. 0.2047470.204747
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s − 13-s + (0.5 + 0.866i)19-s + 0.999·21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s − 43-s + (−0.499 + 0.866i)49-s − 0.999·57-s + (−1 − 1.73i)61-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s − 13-s + (0.5 + 0.866i)19-s + 0.999·21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + (0.5 − 0.866i)39-s − 43-s + (−0.499 + 0.866i)49-s − 0.999·57-s + (−1 − 1.73i)61-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.895−0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s) L(s)=((0.895−0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.895−0.444i0.895 - 0.444i
Analytic conductor: 0.04192140.0419214
Root analytic conductor: 0.2047470.204747
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ84(53,⋅)\chi_{84} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :0), 0.895−0.444i)(2,\ 84,\ (\ :0),\ 0.895 - 0.444i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.45507118420.4550711842
L(12)L(\frac12) ≈\approx 0.45507118420.4550711842
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1+T+T2 1 + T + T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1−2T+T2 1 - 2T + T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−14.74600246740631530719188745074, −13.63570593099300485737171669080, −12.33651690769049716182029767608, −11.28616606617349332842095808165, −10.10977840826758241325303230439, −9.514310179830675798001626081592, −7.74268924650426005669373898019, −6.33268371238830377081639818438, −4.89098280278214127535463544377, −3.52084052442906496218604031148, 2.56497434085969418732679777437, 5.07920460323713432335456360487, 6.30895696473354994681334896900, 7.44725460239841168931737259170, 8.796010299104171695587299127572, 10.11348932277312344731628512468, 11.57081641310592920438165715557, 12.27926998190461195602564601842, 13.20498120399190008275121892141, 14.33197631703146060715751911730

Graph of the ZZ-function along the critical line