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Properties

Label 4-192e2-1.1-c7e2-0-1
Degree 44
Conductor 3686436864
Sign 11
Analytic cond. 3597.353597.35
Root an. cond. 7.744547.74454
Motivic weight 77
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.18e3·9-s − 2.92e4·13-s + 1.56e5·25-s − 5.59e5·37-s − 1.38e6·49-s + 7.07e6·61-s + 1.25e7·73-s + 4.78e6·81-s − 2.44e7·97-s − 3.36e7·109-s + 6.39e7·117-s − 3.89e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.15e8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 9-s − 3.68·13-s + 2·25-s − 1.81·37-s − 1.68·49-s + 3.98·61-s + 3.77·73-s + 81-s − 2.72·97-s − 2.48·109-s + 3.68·117-s − 2·121-s + 8.21·169-s + ⋯

Functional equation

Λ(s)=(36864s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(36864s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3686436864    =    212⋅322^{12} \cdot 3^{2}
Sign: 11
Analytic conductor: 3597.353597.35
Root analytic conductor: 7.744547.74454
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 36864, ( :7/2,7/2), 1)(4,\ 36864,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 0.092177706220.09217770622
L(12)L(\frac12) ≈\approx 0.092177706220.09217770622
L(92)L(\frac{9}{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
3C2C_2 1+p7T2 1 + p^{7} T^{2}
good5C2C_2 (1−p7T2)2 ( 1 - p^{7} T^{2} )^{2}
7C2C_2 (1−508T+p7T2)(1+508T+p7T2) ( 1 - 508 T + p^{7} T^{2} )( 1 + 508 T + p^{7} T^{2} )
11C2C_2 (1+p7T2)2 ( 1 + p^{7} T^{2} )^{2}
13C2C_2 (1+14614T+p7T2)2 ( 1 + 14614 T + p^{7} T^{2} )^{2}
17C2C_2 (1−p7T2)2 ( 1 - p^{7} T^{2} )^{2}
19C2C_2 (1−57448T+p7T2)(1+57448T+p7T2) ( 1 - 57448 T + p^{7} T^{2} )( 1 + 57448 T + p^{7} T^{2} )
23C2C_2 (1+p7T2)2 ( 1 + p^{7} T^{2} )^{2}
29C2C_2 (1−p7T2)2 ( 1 - p^{7} T^{2} )^{2}
31C2C_2 (1−178916T+p7T2)(1+178916T+p7T2) ( 1 - 178916 T + p^{7} T^{2} )( 1 + 178916 T + p^{7} T^{2} )
37C2C_2 (1+279710T+p7T2)2 ( 1 + 279710 T + p^{7} T^{2} )^{2}
41C2C_2 (1−p7T2)2 ( 1 - p^{7} T^{2} )^{2}
43C2C_2 (1−1035224T+p7T2)(1+1035224T+p7T2) ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 1035224 T + p^{7} T^{2} )
47C2C_2 (1+p7T2)2 ( 1 + p^{7} T^{2} )^{2}
53C2C_2 (1−p7T2)2 ( 1 - p^{7} T^{2} )^{2}
59C2C_2 (1+p7T2)2 ( 1 + p^{7} T^{2} )^{2}
61C2C_2 (1−3535546T+p7T2)2 ( 1 - 3535546 T + p^{7} T^{2} )^{2}
67C2C_2 (1−385072T+p7T2)(1+385072T+p7T2) ( 1 - 385072 T + p^{7} T^{2} )( 1 + 385072 T + p^{7} T^{2} )
71C2C_2 (1+p7T2)2 ( 1 + p^{7} T^{2} )^{2}
73C2C_2 (1−6274810T+p7T2)2 ( 1 - 6274810 T + p^{7} T^{2} )^{2}
79C2C_2 (1−8763044T+p7T2)(1+8763044T+p7T2) ( 1 - 8763044 T + p^{7} T^{2} )( 1 + 8763044 T + p^{7} T^{2} )
83C2C_2 (1+p7T2)2 ( 1 + p^{7} T^{2} )^{2}
89C2C_2 (1−p7T2)2 ( 1 - p^{7} T^{2} )^{2}
97C2C_2 (1+12245198T+p7T2)2 ( 1 + 12245198 T + p^{7} T^{2} )^{2}
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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

−12.06307291823413843129222722742, −10.98533914181917178306603362293, −10.52472014436010128017524532382, −9.830200720263015215291814085503, −9.659264011391398292781591133079, −9.117614824531049181911078640840, −8.331233678408138834740670717036, −8.101478860613731474127700911633, −7.31895045093175899519348238932, −6.73375637921156299919770392406, −6.73233544760550273040317385953, −5.31186854528034391930689751108, −5.16904429864291205140409461974, −4.91067308666795361561061468239, −3.88789405713229226942239345080, −3.09753818156138453747627976441, −2.41399973116681595484826347764, −2.28135334293542208660316386341, −1.02502713302394980122508437558, −0.083591046351202198967947105267, 0.083591046351202198967947105267, 1.02502713302394980122508437558, 2.28135334293542208660316386341, 2.41399973116681595484826347764, 3.09753818156138453747627976441, 3.88789405713229226942239345080, 4.91067308666795361561061468239, 5.16904429864291205140409461974, 5.31186854528034391930689751108, 6.73233544760550273040317385953, 6.73375637921156299919770392406, 7.31895045093175899519348238932, 8.101478860613731474127700911633, 8.331233678408138834740670717036, 9.117614824531049181911078640840, 9.659264011391398292781591133079, 9.830200720263015215291814085503, 10.52472014436010128017524532382, 10.98533914181917178306603362293, 12.06307291823413843129222722742

Graph of the ZZ-function along the critical line