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Properties

Label 4-7800e2-1.1-c1e2-0-1
Degree 44
Conductor 6084000060840000
Sign 11
Analytic cond. 3879.213879.21
Root an. cond. 7.891977.89197
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·7-s + 3·9-s − 2·11-s − 2·13-s + 2·17-s − 4·19-s − 4·21-s + 4·23-s − 4·27-s − 6·29-s + 6·31-s + 4·33-s − 4·37-s + 4·39-s − 4·43-s + 10·47-s − 9·49-s − 4·51-s − 6·53-s + 8·57-s − 10·59-s + 2·61-s + 6·63-s + 10·67-s − 8·69-s − 4·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.755·7-s + 9-s − 0.603·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 0.834·23-s − 0.769·27-s − 1.11·29-s + 1.07·31-s + 0.696·33-s − 0.657·37-s + 0.640·39-s − 0.609·43-s + 1.45·47-s − 9/7·49-s − 0.560·51-s − 0.824·53-s + 1.05·57-s − 1.30·59-s + 0.256·61-s + 0.755·63-s + 1.22·67-s − 0.963·69-s − 0.474·71-s + ⋯

Functional equation

Λ(s)=(60840000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(60840000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 60840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6084000060840000    =    26⋅32⋅54⋅1322^{6} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 3879.213879.21
Root analytic conductor: 7.891977.89197
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 60840000, ( :1/2,1/2), 1)(4,\ 60840000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad2 1 1
3C1C_1 (1+T)2 ( 1 + T )^{2}
5 1 1
13C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1−2T+13T2−2pT3+p2T4 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+2T+21T2+2pT3+p2T4 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−2T+27T2−2pT3+p2T4 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4}
19C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
23C4C_4 1−4T+42T2−4pT3+p2T4 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+6T+35T2+6pT3+p2T4 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−6T+69T2−6pT3+p2T4 1 - 6 T + 69 T^{2} - 6 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+4T+46T2+4pT3+p2T4 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
43D4D_{4} 1+4T+18T2+4pT3+p2T4 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−10T+69T2−10pT3+p2T4 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+6T+107T2+6pT3+p2T4 1 + 6 T + 107 T^{2} + 6 p T^{3} + p^{2} T^{4}
59D4D_{4} 1+10T+141T2+10pT3+p2T4 1 + 10 T + 141 T^{2} + 10 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−2T+115T2−2pT3+p2T4 1 - 2 T + 115 T^{2} - 2 p T^{3} + p^{2} T^{4}
67D4D_{4} 1−10T+141T2−10pT3+p2T4 1 - 10 T + 141 T^{2} - 10 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+4T+114T2+4pT3+p2T4 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+4T+22T2+4pT3+p2T4 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+12T+186T2+12pT3+p2T4 1 + 12 T + 186 T^{2} + 12 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−2T+5T2−2pT3+p2T4 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C4C_4 1+4T−90T2+4pT3+p2T4 1 + 4 T - 90 T^{2} + 4 p T^{3} + p^{2} 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

−7.65505552484139893910982204867, −7.34546056978665090010019695468, −6.86364996755545557494333988767, −6.76148596586763014391602417703, −6.10534376852989647463977389456, −6.08531503458998691003173646863, −5.42986288125131486958830049550, −5.29349847439803882073004442338, −4.86700783069264527152267584865, −4.72456304988199802748126408139, −4.09941838692569730906743495863, −4.00923235246730581677128086819, −3.19958270232897897499669005635, −2.99848048223864324548280557079, −2.31899830698031171876791112009, −2.00770662598978189488872407011, −1.32737307812327504819314845613, −1.11390845387100822871916869345, 0, 0, 1.11390845387100822871916869345, 1.32737307812327504819314845613, 2.00770662598978189488872407011, 2.31899830698031171876791112009, 2.99848048223864324548280557079, 3.19958270232897897499669005635, 4.00923235246730581677128086819, 4.09941838692569730906743495863, 4.72456304988199802748126408139, 4.86700783069264527152267584865, 5.29349847439803882073004442338, 5.42986288125131486958830049550, 6.08531503458998691003173646863, 6.10534376852989647463977389456, 6.76148596586763014391602417703, 6.86364996755545557494333988767, 7.34546056978665090010019695468, 7.65505552484139893910982204867

Graph of the ZZ-function along the critical line