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Properties

Label 8-45e4-1.1-c4e4-0-1
Degree 88
Conductor 41006254100625
Sign 11
Analytic cond. 468.195468.195
Root an. cond. 2.156762.15676
Motivic weight 44
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  + 18·4-s − 48·7-s + 808·13-s + 91·16-s − 1.76e3·19-s − 250·25-s − 864·28-s − 1.65e3·31-s + 344·37-s + 5.77e3·43-s + 656·49-s + 1.45e4·52-s − 2.01e4·61-s + 2.05e3·64-s − 1.58e3·67-s + 88·73-s − 3.16e4·76-s + 1.77e4·79-s − 3.87e4·91-s + 2.74e4·97-s − 4.50e3·100-s − 400·103-s − 6.52e3·109-s − 4.36e3·112-s + 4.74e4·121-s − 2.98e4·124-s + 127-s + ⋯
L(s)  = 1  + 9/8·4-s − 0.979·7-s + 4.78·13-s + 0.355·16-s − 4.87·19-s − 2/5·25-s − 1.10·28-s − 1.72·31-s + 0.251·37-s + 3.12·43-s + 0.273·49-s + 5.37·52-s − 5.42·61-s + 0.500·64-s − 0.352·67-s + 0.0165·73-s − 5.48·76-s + 2.84·79-s − 4.68·91-s + 2.91·97-s − 0.449·100-s − 0.0377·103-s − 0.548·109-s − 0.348·112-s + 3.24·121-s − 1.93·124-s + 6.20e−5·127-s + ⋯

Functional equation

Λ(s)=(4100625s/2 ΓC(s)4 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(4100625s/2 ΓC(s+2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 41006254100625    =    38⋅543^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 468.195468.195
Root analytic conductor: 2.156762.15676
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 4100625, ( :2,2,2,2), 1)(8,\ 4100625,\ (\ :2, 2, 2, 2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.8819544312.881954431
L(12)L(\frac12) ≈\approx 2.8819544312.881954431
L(3)L(3) 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)
bad3 1 1
5C2C_2 (1+p3T2)2 ( 1 + p^{3} T^{2} )^{2}
good2D4×C2D_4\times C_2 1−9pT2+233T4−9p9T6+p16T8 1 - 9 p T^{2} + 233 T^{4} - 9 p^{9} T^{6} + p^{16} T^{8}
7D4D_{4} (1+24T+536T2+24p4T3+p8T4)2 ( 1 + 24 T + 536 T^{2} + 24 p^{4} T^{3} + p^{8} T^{4} )^{2}
11D4×C2D_4\times C_2 1−47448T2+962099378T4−47448p8T6+p16T8 1 - 47448 T^{2} + 962099378 T^{4} - 47448 p^{8} T^{6} + p^{16} T^{8}
13D4D_{4} (1−404T+95676T2−404p4T3+p8T4)2 ( 1 - 404 T + 95676 T^{2} - 404 p^{4} T^{3} + p^{8} T^{4} )^{2}
17D4×C2D_4\times C_2 1−138948T2+10035485318T4−138948p8T6+p16T8 1 - 138948 T^{2} + 10035485318 T^{4} - 138948 p^{8} T^{6} + p^{16} T^{8}
19D4D_{4} (1+880T+453882T2+880p4T3+p8T4)2 ( 1 + 880 T + 453882 T^{2} + 880 p^{4} T^{3} + p^{8} T^{4} )^{2}
23D4×C2D_4\times C_2 1−254860T2+172735907622T4−254860p8T6+p16T8 1 - 254860 T^{2} + 172735907622 T^{4} - 254860 p^{8} T^{6} + p^{16} T^{8}
29D4×C2D_4\times C_2 1−1417500T2+1274204988422T4−1417500p8T6+p16T8 1 - 1417500 T^{2} + 1274204988422 T^{4} - 1417500 p^{8} T^{6} + p^{16} T^{8}
31D4D_{4} (1+828T+848798T2+828p4T3+p8T4)2 ( 1 + 828 T + 848798 T^{2} + 828 p^{4} T^{3} + p^{8} T^{4} )^{2}
37D4D_{4} (1−172T+2066508T2−172p4T3+p8T4)2 ( 1 - 172 T + 2066508 T^{2} - 172 p^{4} T^{3} + p^{8} T^{4} )^{2}
41D4×C2D_4\times C_2 1−4942080T2+12337515177602T4−4942080p8T6+p16T8 1 - 4942080 T^{2} + 12337515177602 T^{4} - 4942080 p^{8} T^{6} + p^{16} T^{8}
43D4D_{4} (1−2888T+8193738T2−2888p4T3+p8T4)2 ( 1 - 2888 T + 8193738 T^{2} - 2888 p^{4} T^{3} + p^{8} T^{4} )^{2}
47D4×C2D_4\times C_2 1−12953940T2+88587954654182T4−12953940p8T6+p16T8 1 - 12953940 T^{2} + 88587954654182 T^{4} - 12953940 p^{8} T^{6} + p^{16} T^{8}
53D4×C2D_4\times C_2 1+4980300T2+23234867625062T4+4980300p8T6+p16T8 1 + 4980300 T^{2} + 23234867625062 T^{4} + 4980300 p^{8} T^{6} + p^{16} T^{8}
59D4×C2D_4\times C_2 1−26408088T2+396550023159218T4−26408088p8T6+p16T8 1 - 26408088 T^{2} + 396550023159218 T^{4} - 26408088 p^{8} T^{6} + p^{16} T^{8}
61D4D_{4} (1+10084T+53084286T2+10084p4T3+p8T4)2 ( 1 + 10084 T + 53084286 T^{2} + 10084 p^{4} T^{3} + p^{8} T^{4} )^{2}
67D4D_{4} (1+792T+37671218T2+792p4T3+p8T4)2 ( 1 + 792 T + 37671218 T^{2} + 792 p^{4} T^{3} + p^{8} T^{4} )^{2}
71D4×C2D_4\times C_2 1−71497380T2+2526354220369862T4−71497380p8T6+p16T8 1 - 71497380 T^{2} + 2526354220369862 T^{4} - 71497380 p^{8} T^{6} + p^{16} T^{8}
73D4D_{4} (1−44T+42825726T2−44p4T3+p8T4)2 ( 1 - 44 T + 42825726 T^{2} - 44 p^{4} T^{3} + p^{8} T^{4} )^{2}
79D4D_{4} (1−8868T+95944478T2−8868p4T3+p8T4)2 ( 1 - 8868 T + 95944478 T^{2} - 8868 p^{4} T^{3} + p^{8} T^{4} )^{2}
83D4×C2D_4\times C_2 1−181435060T2+12718407730707942T4−181435060p8T6+p16T8 1 - 181435060 T^{2} + 12718407730707942 T^{4} - 181435060 p^{8} T^{6} + p^{16} T^{8}
89D4×C2D_4\times C_2 1−97686688T2+9800859843406338T4−97686688p8T6+p16T8 1 - 97686688 T^{2} + 9800859843406338 T^{4} - 97686688 p^{8} T^{6} + p^{16} T^{8}
97D4D_{4} (1−13708T+217279038T2−13708p4T3+p8T4)2 ( 1 - 13708 T + 217279038 T^{2} - 13708 p^{4} T^{3} + p^{8} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.84803695825375675892535573059, −10.80693544336201295475119994632, −10.77058423902345769162767995327, −10.59240667688862522636706718877, −9.724306903103296351305786888932, −9.300108471976717138844016641575, −8.778761291732889550317254180504, −8.776338765626702950352932583335, −8.691358673445755338811491292137, −8.101466644287967689533201571444, −7.64198423100703328971210906800, −7.34236257637943697151803498539, −6.52235222721893519276187410121, −6.39558972005466015303304961855, −6.17660358491420999031001506089, −6.04914718707421195970427716632, −5.83689647519327057887894640108, −4.60624755953128935111878367634, −4.19251115211408132906753895843, −3.74349138203140334101296640408, −3.59308342954941308091646811226, −2.77067047971433933354780975817, −1.92090961219450480885373930013, −1.70368117018456975846546639600, −0.56448035647332114981281378946, 0.56448035647332114981281378946, 1.70368117018456975846546639600, 1.92090961219450480885373930013, 2.77067047971433933354780975817, 3.59308342954941308091646811226, 3.74349138203140334101296640408, 4.19251115211408132906753895843, 4.60624755953128935111878367634, 5.83689647519327057887894640108, 6.04914718707421195970427716632, 6.17660358491420999031001506089, 6.39558972005466015303304961855, 6.52235222721893519276187410121, 7.34236257637943697151803498539, 7.64198423100703328971210906800, 8.101466644287967689533201571444, 8.691358673445755338811491292137, 8.776338765626702950352932583335, 8.778761291732889550317254180504, 9.300108471976717138844016641575, 9.724306903103296351305786888932, 10.59240667688862522636706718877, 10.77058423902345769162767995327, 10.80693544336201295475119994632, 10.84803695825375675892535573059

Graph of the ZZ-function along the critical line