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Properties

Label 4-189e2-1.1-c3e2-0-6
Degree 44
Conductor 3572135721
Sign 11
Analytic cond. 124.352124.352
Root an. cond. 3.339363.33936
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 14·4-s + 6·5-s + 14·7-s + 36·10-s + 48·11-s + 52·13-s + 84·14-s − 84·16-s + 30·17-s + 64·19-s + 84·20-s + 288·22-s + 60·23-s − 175·25-s + 312·26-s + 196·28-s + 360·29-s − 140·31-s − 216·32-s + 180·34-s + 84·35-s − 230·37-s + 384·38-s + 234·41-s − 938·43-s + 672·44-s + ⋯
L(s)  = 1  + 2.12·2-s + 7/4·4-s + 0.536·5-s + 0.755·7-s + 1.13·10-s + 1.31·11-s + 1.10·13-s + 1.60·14-s − 1.31·16-s + 0.428·17-s + 0.772·19-s + 0.939·20-s + 2.79·22-s + 0.543·23-s − 7/5·25-s + 2.35·26-s + 1.32·28-s + 2.30·29-s − 0.811·31-s − 1.19·32-s + 0.907·34-s + 0.405·35-s − 1.02·37-s + 1.63·38-s + 0.891·41-s − 3.32·43-s + 2.30·44-s + ⋯

Functional equation

Λ(s)=(35721s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(35721s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3572135721    =    36⋅723^{6} \cdot 7^{2}
Sign: 11
Analytic conductor: 124.352124.352
Root analytic conductor: 3.339363.33936
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 35721, ( :3/2,3/2), 1)(4,\ 35721,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 9.3767238929.376723892
L(12)L(\frac12) ≈\approx 9.3767238929.376723892
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
7C1C_1 (1−pT)2 ( 1 - p T )^{2}
good2D4D_{4} 1−3pT+11pT2−3p4T3+p6T4 1 - 3 p T + 11 p T^{2} - 3 p^{4} T^{3} + p^{6} T^{4}
5D4D_{4} 1−6T+211T2−6p3T3+p6T4 1 - 6 T + 211 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1−48T+2470T2−48p3T3+p6T4 1 - 48 T + 2470 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−4pT+3342T2−4p4T3+p6T4 1 - 4 p T + 3342 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4}
17D4D_{4} 1−30T+7699T2−30p3T3+p6T4 1 - 30 T + 7699 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−64T+3942T2−64p3T3+p6T4 1 - 64 T + 3942 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−60T+494pT2−60p3T3+p6T4 1 - 60 T + 494 p T^{2} - 60 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−360T+77290T2−360p3T3+p6T4 1 - 360 T + 77290 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+140T+48930T2+140p3T3+p6T4 1 + 140 T + 48930 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+230T+98979T2+230p3T3+p6T4 1 + 230 T + 98979 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−234T+26683T2−234p3T3+p6T4 1 - 234 T + 26683 T^{2} - 234 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+938T+378543T2+938p3T3+p6T4 1 + 938 T + 378543 T^{2} + 938 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1−618T+210199T2−618p3T3+p6T4 1 - 618 T + 210199 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+420T+64606T2+420p3T3+p6T4 1 + 420 T + 64606 T^{2} + 420 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1−282T+411439T2−282p3T3+p6T4 1 - 282 T + 411439 T^{2} - 282 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+32T+205386T2+32p3T3+p6T4 1 + 32 T + 205386 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−544T+535542T2−544p3T3+p6T4 1 - 544 T + 535542 T^{2} - 544 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+504T+736126T2+504p3T3+p6T4 1 + 504 T + 736126 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+764T+653958T2+764p3T3+p6T4 1 + 764 T + 653958 T^{2} + 764 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−238T+125439T2−238p3T3+p6T4 1 - 238 T + 125439 T^{2} - 238 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+522T+651823T2+522p3T3+p6T4 1 + 522 T + 651823 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−708T+1163542T2−708p3T3+p6T4 1 - 708 T + 1163542 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−664T+1779618T2−664p3T3+p6T4 1 - 664 T + 1779618 T^{2} - 664 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.27632987659299655646781178799, −12.11050418120095162927090243564, −11.53596273464521670832837473914, −11.38877843987926079005848866597, −10.39527046502980608642461348633, −10.11273323659505284049996330443, −9.265470538002553362706371216565, −8.883230783940692852695745409819, −8.342258761951479821104636953239, −7.64659685835112797103145769696, −6.73961172359149170084649142515, −6.44846109697459173396428217355, −5.62749816620686311188033767704, −5.51343900559437254962157215947, −4.66444943384070729710683771265, −4.30097411355325161304113297813, −3.48890578831053268065585151651, −3.22541706504688024702033440526, −1.92515513340998566052728157358, −1.10424273914690404018298632997, 1.10424273914690404018298632997, 1.92515513340998566052728157358, 3.22541706504688024702033440526, 3.48890578831053268065585151651, 4.30097411355325161304113297813, 4.66444943384070729710683771265, 5.51343900559437254962157215947, 5.62749816620686311188033767704, 6.44846109697459173396428217355, 6.73961172359149170084649142515, 7.64659685835112797103145769696, 8.342258761951479821104636953239, 8.883230783940692852695745409819, 9.265470538002553362706371216565, 10.11273323659505284049996330443, 10.39527046502980608642461348633, 11.38877843987926079005848866597, 11.53596273464521670832837473914, 12.11050418120095162927090243564, 12.27632987659299655646781178799

Graph of the ZZ-function along the critical line