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Properties

Label 2-432-12.11-c3-0-1
Degree 22
Conductor 432432
Sign −1-1
Analytic cond. 25.488825.4888
Root an. cond. 5.048645.04864
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12.3i·5-s + 21.4i·7-s − 15.5·11-s − 2·13-s + 24.7i·17-s + 85.6i·19-s − 124.·23-s − 27.9·25-s − 272. i·29-s − 278. i·31-s − 265.·35-s + 128·37-s + 296. i·41-s − 42.8i·43-s − 592.·47-s + ⋯
L(s)  = 1  + 1.10i·5-s + 1.15i·7-s − 0.427·11-s − 0.0426·13-s + 0.352i·17-s + 1.03i·19-s − 1.13·23-s − 0.223·25-s − 1.74i·29-s − 1.61i·31-s − 1.27·35-s + 0.568·37-s + 1.13i·41-s − 0.151i·43-s − 1.83·47-s + ⋯

Functional equation

Λ(s)=(432s/2 ΓC(s) L(s)=(− Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(432s/2 ΓC(s+3/2) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 432432    =    24⋅332^{4} \cdot 3^{3}
Sign: −1-1
Analytic conductor: 25.488825.4888
Root analytic conductor: 5.048645.04864
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ432(431,⋅)\chi_{432} (431, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 432, ( :3/2), −1)(2,\ 432,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) ≈\approx 0.89148413340.8914841334
L(12)L(\frac12) ≈\approx 0.89148413340.8914841334
L(52)L(\frac{5}{2}) 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 1
good5 1−12.3iT−125T2 1 - 12.3iT - 125T^{2}
7 1−21.4iT−343T2 1 - 21.4iT - 343T^{2}
11 1+15.5T+1.33e3T2 1 + 15.5T + 1.33e3T^{2}
13 1+2T+2.19e3T2 1 + 2T + 2.19e3T^{2}
17 1−24.7iT−4.91e3T2 1 - 24.7iT - 4.91e3T^{2}
19 1−85.6iT−6.85e3T2 1 - 85.6iT - 6.85e3T^{2}
23 1+124.T+1.21e4T2 1 + 124.T + 1.21e4T^{2}
29 1+272.iT−2.43e4T2 1 + 272. iT - 2.43e4T^{2}
31 1+278.iT−2.97e4T2 1 + 278. iT - 2.97e4T^{2}
37 1−128T+5.06e4T2 1 - 128T + 5.06e4T^{2}
41 1−296.iT−6.89e4T2 1 - 296. iT - 6.89e4T^{2}
43 1+42.8iT−7.95e4T2 1 + 42.8iT - 7.95e4T^{2}
47 1+592.T+1.03e5T2 1 + 592.T + 1.03e5T^{2}
53 1−259.iT−1.48e5T2 1 - 259. iT - 1.48e5T^{2}
59 1+530.T+2.05e5T2 1 + 530.T + 2.05e5T^{2}
61 1+340T+2.26e5T2 1 + 340T + 2.26e5T^{2}
67 1−899.iT−3.00e5T2 1 - 899. iT - 3.00e5T^{2}
71 1+966.T+3.57e5T2 1 + 966.T + 3.57e5T^{2}
73 1−817T+3.89e5T2 1 - 817T + 3.89e5T^{2}
79 1−214.iT−4.93e5T2 1 - 214. iT - 4.93e5T^{2}
83 1+358.T+5.71e5T2 1 + 358.T + 5.71e5T^{2}
89 1+915.iT−7.04e5T2 1 + 915. iT - 7.04e5T^{2}
97 1+965T+9.12e5T2 1 + 965T + 9.12e5T^{2}
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   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

−11.29010122376503711494913806426, −10.18263750497562831447497832589, −9.605597375827719697380407576055, −8.271439012810719172821251288729, −7.65787693277461397883305921115, −6.22002079000557253059737668710, −5.85142592237865463173727599084, −4.27620730882754556140098380650, −2.96563156863142463859765261970, −2.06154878611918138817789955728, 0.28472710602980690226419601398, 1.49347763875534544180407815435, 3.27474073236677394853512132858, 4.56293349876928932537543826528, 5.17523392849378332063202812408, 6.64403406343019837191535275126, 7.53019044534645123921506682827, 8.505649414406756092502055260485, 9.313448174070513464642292375040, 10.34437640417272867410988132201

Graph of the ZZ-function along the critical line