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Properties

Label 4-1216e2-1.1-c1e2-0-7
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
Motivic weight 11
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  + 3-s − 8·7-s + 3·9-s − 6·11-s + 2·13-s + 6·17-s + 7·19-s − 8·21-s + 6·23-s + 5·25-s + 8·27-s + 4·31-s − 6·33-s + 20·37-s + 2·39-s − 9·41-s − 4·43-s + 34·49-s + 6·51-s + 6·53-s + 7·57-s − 9·59-s − 4·61-s − 24·63-s − 7·67-s + 6·69-s + 6·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 3.02·7-s + 9-s − 1.80·11-s + 0.554·13-s + 1.45·17-s + 1.60·19-s − 1.74·21-s + 1.25·23-s + 25-s + 1.53·27-s + 0.718·31-s − 1.04·33-s + 3.28·37-s + 0.320·39-s − 1.40·41-s − 0.609·43-s + 34/7·49-s + 0.840·51-s + 0.824·53-s + 0.927·57-s − 1.17·59-s − 0.512·61-s − 3.02·63-s − 0.855·67-s + 0.722·69-s + 0.712·71-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9934929831.993492983
L(12)L(\frac12) ≈\approx 1.9934929831.993492983
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
19C2C_2 1−7T+pT2 1 - 7 T + p T^{2}
good3C22C_2^2 1−T−2T2−pT3+p2T4 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4}
5C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
7C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
11C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
13C2C_2 (1−7T+pT2)(1+5T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} )
17C22C_2^2 1−6T+19T2−6pT3+p2T4 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−6T+13T2−6pT3+p2T4 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
31C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
37C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
41C22C_2^2 1+9T+40T2+9pT3+p2T4 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4}
43C22C_2^2 1+4T−27T2+4pT3+p2T4 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C22C_2^2 1−6T−17T2−6pT3+p2T4 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+9T+22T2+9pT3+p2T4 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+4T−45T2+4pT3+p2T4 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+7T−18T2+7pT3+p2T4 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−6T−35T2−6pT3+p2T4 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4}
73C22C_2^2 1−T−72T2−pT3+p2T4 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4}
79C2C_2 (1−17T+pT2)(1+13T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} )
83C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
89C22C_2^2 1+6T−53T2+6pT3+p2T4 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4}
97C22C_2^2 1+17T+192T2+17pT3+p2T4 1 + 17 T + 192 T^{2} + 17 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

−9.892050146137477846198667778016, −9.648076560967476764093598862214, −9.248943792876180282106526506103, −8.888678574573461179085764885642, −8.204052810644402587360936798804, −7.900225712865470621351691421551, −7.44424171897927928764187474818, −7.05006633339860145427753795746, −6.63394717743279723533903728594, −6.35575204356113544667431827066, −5.65396617700276012431969186488, −5.48198430761843276797236102550, −4.76468475562860726325209260769, −4.29876058044247304345007542289, −3.35407486990990152512571686815, −3.29096627504812227289805055781, −2.78225318144012742940223962659, −2.72847360440711388628494488929, −1.21483248383309752623603914887, −0.66564732616554931686820538104, 0.66564732616554931686820538104, 1.21483248383309752623603914887, 2.72847360440711388628494488929, 2.78225318144012742940223962659, 3.29096627504812227289805055781, 3.35407486990990152512571686815, 4.29876058044247304345007542289, 4.76468475562860726325209260769, 5.48198430761843276797236102550, 5.65396617700276012431969186488, 6.35575204356113544667431827066, 6.63394717743279723533903728594, 7.05006633339860145427753795746, 7.44424171897927928764187474818, 7.900225712865470621351691421551, 8.204052810644402587360936798804, 8.888678574573461179085764885642, 9.248943792876180282106526506103, 9.648076560967476764093598862214, 9.892050146137477846198667778016

Graph of the ZZ-function along the critical line