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Properties

Label 4-28e4-1.1-c5e2-0-7
Degree 44
Conductor 614656614656
Sign 11
Analytic cond. 15810.715810.7
Root an. cond. 11.213411.2134
Motivic weight 55
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  + 14·3-s − 42·5-s + 181·9-s + 294·11-s − 140·13-s − 588·15-s + 1.30e3·17-s + 1.44e3·19-s − 2.64e3·23-s − 4.92e3·25-s + 5.72e3·27-s + 1.66e3·29-s + 1.47e4·31-s + 4.11e3·33-s + 5.18e3·37-s − 1.96e3·39-s − 5.12e3·41-s + 4.52e3·43-s − 7.60e3·45-s + 1.49e4·47-s + 1.82e4·51-s + 2.40e4·53-s − 1.23e4·55-s + 2.01e4·57-s − 3.88e4·59-s − 2.36e4·61-s + 5.88e3·65-s + ⋯
L(s)  = 1  + 0.898·3-s − 0.751·5-s + 0.744·9-s + 0.732·11-s − 0.229·13-s − 0.674·15-s + 1.09·17-s + 0.916·19-s − 1.04·23-s − 1.57·25-s + 1.51·27-s + 0.368·29-s + 2.76·31-s + 0.657·33-s + 0.622·37-s − 0.206·39-s − 0.476·41-s + 0.372·43-s − 0.559·45-s + 0.990·47-s + 0.981·51-s + 1.17·53-s − 0.550·55-s + 0.823·57-s − 1.45·59-s − 0.812·61-s + 0.172·65-s + ⋯

Functional equation

Λ(s)=(614656s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(614656s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 614656614656    =    28⋅742^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 15810.715810.7
Root analytic conductor: 11.213411.2134
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 614656, ( :5/2,5/2), 1)(4,\ 614656,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 5.8170485785.817048578
L(12)L(\frac12) ≈\approx 5.8170485785.817048578
L(72)L(\frac{7}{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
7 1 1
good3D4D_{4} 1−14T+5pT2−14p5T3+p10T4 1 - 14 T + 5 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4}
5C2C_2 (1+21T+p5T2)2 ( 1 + 21 T + p^{5} T^{2} )^{2}
11D4D_{4} 1−294T+114391T2−294p5T3+p10T4 1 - 294 T + 114391 T^{2} - 294 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+140T+728766T2+140p5T3+p10T4 1 + 140 T + 728766 T^{2} + 140 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1−1302T+3095035T2−1302p5T3+p10T4 1 - 1302 T + 3095035 T^{2} - 1302 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1−1442T+4119519T2−1442p5T3+p10T4 1 - 1442 T + 4119519 T^{2} - 1442 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+2646T+8890015T2+2646p5T3+p10T4 1 + 2646 T + 8890015 T^{2} + 2646 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1−1668T+40800574T2−1668p5T3+p10T4 1 - 1668 T + 40800574 T^{2} - 1668 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1−14798T+109078503T2−14798p5T3+p10T4 1 - 14798 T + 109078503 T^{2} - 14798 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1−5182T+71101515T2−5182p5T3+p10T4 1 - 5182 T + 71101515 T^{2} - 5182 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+5124T+23950966T2+5124p5T3+p10T4 1 + 5124 T + 23950966 T^{2} + 5124 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1−4520T+240418566T2−4520p5T3+p10T4 1 - 4520 T + 240418566 T^{2} - 4520 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1−14994T+427056103T2−14994p5T3+p10T4 1 - 14994 T + 427056103 T^{2} - 14994 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1−24006T+935516275T2−24006p5T3+p10T4 1 - 24006 T + 935516275 T^{2} - 24006 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+38850T+1324947343T2+38850p5T3+p10T4 1 + 38850 T + 1324947343 T^{2} + 38850 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+23618T+1734277563T2+23618p5T3+p10T4 1 + 23618 T + 1734277563 T^{2} + 23618 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+32002T+1139838495T2+32002p5T3+p10T4 1 + 32002 T + 1139838495 T^{2} + 32002 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1−89376T+5425689166T2−89376p5T3+p10T4 1 - 89376 T + 5425689166 T^{2} - 89376 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1+47138T+4432072947T2+47138p5T3+p10T4 1 + 47138 T + 4432072947 T^{2} + 47138 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1−40970T+6023150703T2−40970p5T3+p10T4 1 - 40970 T + 6023150703 T^{2} - 40970 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1+68376T+8908447510T2+68376p5T3+p10T4 1 + 68376 T + 8908447510 T^{2} + 68376 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1−123102T+13551221779T2−123102p5T3+p10T4 1 - 123102 T + 13551221779 T^{2} - 123102 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+43652T+8867162790T2+43652p5T3+p10T4 1 + 43652 T + 8867162790 T^{2} + 43652 p^{5} T^{3} + p^{10} 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

−9.781033030044834136119114271841, −9.386328555661137466794587455250, −8.660911861215947246859385904483, −8.629000350234050557449025953754, −7.80542821429889573465511720678, −7.76291644811865227322924008998, −7.49475661433990486094304128338, −6.76167899013312170564189480595, −6.19480858536123440781376662154, −6.03557594406199370798544250606, −5.20324449079067706237171438206, −4.68266917979253238955802365286, −4.15584153152647268606535579632, −3.90839344006863622562427890211, −3.21741337691319608164364491180, −2.90364524360692666053296665477, −2.25744959001926327203080281883, −1.58262588482208672471368590753, −0.927685826378288273855472556890, −0.56260882415400124601759883494, 0.56260882415400124601759883494, 0.927685826378288273855472556890, 1.58262588482208672471368590753, 2.25744959001926327203080281883, 2.90364524360692666053296665477, 3.21741337691319608164364491180, 3.90839344006863622562427890211, 4.15584153152647268606535579632, 4.68266917979253238955802365286, 5.20324449079067706237171438206, 6.03557594406199370798544250606, 6.19480858536123440781376662154, 6.76167899013312170564189480595, 7.49475661433990486094304128338, 7.76291644811865227322924008998, 7.80542821429889573465511720678, 8.629000350234050557449025953754, 8.660911861215947246859385904483, 9.386328555661137466794587455250, 9.781033030044834136119114271841

Graph of the ZZ-function along the critical line