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Properties

Label 4-14e4-1.1-c2e2-0-4
Degree 44
Conductor 3841638416
Sign 11
Analytic cond. 28.522128.5221
Root an. cond. 2.310972.31097
Motivic weight 22
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  + 4·2-s + 12·4-s + 32·8-s + 18·9-s + 80·16-s + 72·18-s − 48·25-s − 80·29-s + 192·32-s + 216·36-s − 48·37-s − 192·50-s − 180·53-s − 320·58-s + 448·64-s + 576·72-s − 192·74-s + 243·81-s − 576·100-s − 720·106-s + 240·109-s + 60·113-s − 960·116-s + 242·121-s + 127-s + 1.02e3·128-s + 131-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 2·9-s + 5·16-s + 4·18-s − 1.91·25-s − 2.75·29-s + 6·32-s + 6·36-s − 1.29·37-s − 3.83·50-s − 3.39·53-s − 5.51·58-s + 7·64-s + 8·72-s − 2.59·74-s + 3·81-s − 5.75·100-s − 6.79·106-s + 2.20·109-s + 0.530·113-s − 8.27·116-s + 2·121-s + 0.00787·127-s + 8·128-s + 0.00763·131-s + ⋯

Functional equation

Λ(s)=(38416s/2 ΓC(s)2 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(38416s/2 ΓC(s+1)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3841638416    =    24⋅742^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 28.522128.5221
Root analytic conductor: 2.310972.31097
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 38416, ( :1,1), 1)(4,\ 38416,\ (\ :1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 8.4688976058.468897605
L(12)L(\frac12) ≈\approx 8.4688976058.468897605
L(2)L(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)
bad2C1C_1 (1−pT)2 ( 1 - p T )^{2}
7 1 1
good3C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
5C22C_2^2 1+48T2+p4T4 1 + 48 T^{2} + p^{4} T^{4}
11C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
13C22C_2^2 1−240T2+p4T4 1 - 240 T^{2} + p^{4} T^{4}
17C22C_2^2 1−480T2+p4T4 1 - 480 T^{2} + p^{4} T^{4}
19C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
23C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
29C2C_2 (1+40T+p2T2)2 ( 1 + 40 T + p^{2} T^{2} )^{2}
31C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
37C2C_2 (1+24T+p2T2)2 ( 1 + 24 T + p^{2} T^{2} )^{2}
41C22C_2^2 1+1440T2+p4T4 1 + 1440 T^{2} + p^{4} T^{4}
43C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
47C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
53C2C_2 (1+90T+p2T2)2 ( 1 + 90 T + p^{2} T^{2} )^{2}
59C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
61C22C_2^2 1−2640T2+p4T4 1 - 2640 T^{2} + p^{4} T^{4}
67C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
71C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
73C22C_2^2 1−10560T2+p4T4 1 - 10560 T^{2} + p^{4} T^{4}
79C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
83C1C_1×\timesC1C_1 (1−pT)2(1+pT)2 ( 1 - p T )^{2}( 1 + p T )^{2}
89C22C_2^2 1+12480T2+p4T4 1 + 12480 T^{2} + p^{4} T^{4}
97C22C_2^2 1−18720T2+p4T4 1 - 18720 T^{2} + p^{4} 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

−12.50253558320000205952189427732, −12.39654354452379733516005459226, −11.66190909579143122276434895134, −11.20082057830641140824196950599, −10.84251826046443850059361702498, −10.19160565300150037263457094531, −9.754399874224431765245108184445, −9.317593332496501984542966825463, −7.999456689505247356627254791528, −7.73080274000703704474193975056, −7.12699510349989248123107152531, −6.85764733959052772624458418871, −5.97508869433613277027620250328, −5.70540597267753828594049404984, −4.78490339785699574423294206504, −4.46897164182764683599057966275, −3.65858846889893197942071872472, −3.43581985813052847828662029175, −1.86967472601457373171388787514, −1.79550254369905278159745837884, 1.79550254369905278159745837884, 1.86967472601457373171388787514, 3.43581985813052847828662029175, 3.65858846889893197942071872472, 4.46897164182764683599057966275, 4.78490339785699574423294206504, 5.70540597267753828594049404984, 5.97508869433613277027620250328, 6.85764733959052772624458418871, 7.12699510349989248123107152531, 7.73080274000703704474193975056, 7.999456689505247356627254791528, 9.317593332496501984542966825463, 9.754399874224431765245108184445, 10.19160565300150037263457094531, 10.84251826046443850059361702498, 11.20082057830641140824196950599, 11.66190909579143122276434895134, 12.39654354452379733516005459226, 12.50253558320000205952189427732

Graph of the ZZ-function along the critical line