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Properties

Label 4-882e2-1.1-c5e2-0-1
Degree 44
Conductor 777924777924
Sign 11
Analytic cond. 20010.520010.5
Root an. cond. 11.893611.8936
Motivic weight 55
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  − 8·2-s + 48·4-s + 53·5-s − 256·8-s − 424·10-s − 191·11-s − 379·13-s + 1.28e3·16-s + 340·17-s − 1.76e3·19-s + 2.54e3·20-s + 1.52e3·22-s − 3.23e3·23-s − 1.74e3·25-s + 3.03e3·26-s − 4.45e3·29-s + 1.99e3·31-s − 6.14e3·32-s − 2.72e3·34-s + 2.05e4·37-s + 1.41e4·38-s − 1.35e4·40-s − 8.81e3·41-s + 1.58e4·43-s − 9.16e3·44-s + 2.58e4·46-s − 3.39e4·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.948·5-s − 1.41·8-s − 1.34·10-s − 0.475·11-s − 0.621·13-s + 5/4·16-s + 0.285·17-s − 1.12·19-s + 1.42·20-s + 0.673·22-s − 1.27·23-s − 0.557·25-s + 0.879·26-s − 0.984·29-s + 0.372·31-s − 1.06·32-s − 0.403·34-s + 2.47·37-s + 1.58·38-s − 1.34·40-s − 0.818·41-s + 1.30·43-s − 0.713·44-s + 1.80·46-s − 2.23·47-s + ⋯

Functional equation

Λ(s)=(777924s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(777924s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 777924777924    =    22⋅34⋅742^{2} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 20010.520010.5
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 777924, ( :5/2,5/2), 1)(4,\ 777924,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.97895041000.9789504100
L(12)L(\frac12) ≈\approx 0.97895041000.9789504100
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)
bad2C1C_1 (1+p2T)2 ( 1 + p^{2} T )^{2}
3 1 1
7 1 1
good5D4D_{4} 1−53T+4552T2−53p5T3+p10T4 1 - 53 T + 4552 T^{2} - 53 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1+191T+24656pT2+191p5T3+p10T4 1 + 191 T + 24656 p T^{2} + 191 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1+379T+488066T2+379p5T3+p10T4 1 + 379 T + 488066 T^{2} + 379 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1−20pT+1908514T2−20p6T3+p10T4 1 - 20 p T + 1908514 T^{2} - 20 p^{6} T^{3} + p^{10} T^{4}
19D4D_{4} 1+1769T+2083758T2+1769p5T3+p10T4 1 + 1769 T + 2083758 T^{2} + 1769 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+3236T+15452206T2+3236p5T3+p10T4 1 + 3236 T + 15452206 T^{2} + 3236 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+4459T+37061638T2+4459p5T3+p10T4 1 + 4459 T + 37061638 T^{2} + 4459 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 1−1994T−6304813T2−1994p5T3+p10T4 1 - 1994 T - 6304813 T^{2} - 1994 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1−20587T+238401006T2−20587p5T3+p10T4 1 - 20587 T + 238401006 T^{2} - 20587 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+8814T+240678562T2+8814p5T3+p10T4 1 + 8814 T + 240678562 T^{2} + 8814 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1−15853T+338678796T2−15853p5T3+p10T4 1 - 15853 T + 338678796 T^{2} - 15853 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1+33912T+687783466T2+33912p5T3+p10T4 1 + 33912 T + 687783466 T^{2} + 33912 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1+49239T+1320998110T2+49239p5T3+p10T4 1 + 49239 T + 1320998110 T^{2} + 49239 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1−56735T+2230528834T2−56735p5T3+p10T4 1 - 56735 T + 2230528834 T^{2} - 56735 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1−67508T+2826067262T2−67508p5T3+p10T4 1 - 67508 T + 2826067262 T^{2} - 67508 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1−75723T+3861149404T2−75723p5T3+p10T4 1 - 75723 T + 3861149404 T^{2} - 75723 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1−8992T−681216182T2−8992p5T3+p10T4 1 - 8992 T - 681216182 T^{2} - 8992 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1+3201T+2311013380T2+3201p5T3+p10T4 1 + 3201 T + 2311013380 T^{2} + 3201 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1−26612T+3997648985T2−26612p5T3+p10T4 1 - 26612 T + 3997648985 T^{2} - 26612 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1−949T+367057696T2−949p5T3+p10T4 1 - 949 T + 367057696 T^{2} - 949 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1+176562T+18855802834T2+176562p5T3+p10T4 1 + 176562 T + 18855802834 T^{2} + 176562 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+129423T+11942811256T2+129423p5T3+p10T4 1 + 129423 T + 11942811256 T^{2} + 129423 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

−9.593686679517239067262318382131, −9.588284496206203999313133115386, −8.544752750726073573826070796095, −8.488991103176270348195720063575, −7.998780579032255700507522745134, −7.63180831823893626339346397335, −7.24336026099987133946932629267, −6.47022427341208347137695788168, −6.33179203700402331810076632712, −5.97897514565038499076907754382, −5.21970523094121470338683659920, −5.08080512159538745278207114790, −3.93942948883886182771147621814, −3.92487015276632896605164943347, −2.72970349279722709516312789864, −2.58498643910345095811980946115, −1.83994450963448474943187488894, −1.76631922511798681376834421417, −0.78171304658874428420891550419, −0.30663075236324279734113369871, 0.30663075236324279734113369871, 0.78171304658874428420891550419, 1.76631922511798681376834421417, 1.83994450963448474943187488894, 2.58498643910345095811980946115, 2.72970349279722709516312789864, 3.92487015276632896605164943347, 3.93942948883886182771147621814, 5.08080512159538745278207114790, 5.21970523094121470338683659920, 5.97897514565038499076907754382, 6.33179203700402331810076632712, 6.47022427341208347137695788168, 7.24336026099987133946932629267, 7.63180831823893626339346397335, 7.998780579032255700507522745134, 8.488991103176270348195720063575, 8.544752750726073573826070796095, 9.588284496206203999313133115386, 9.593686679517239067262318382131

Graph of the ZZ-function along the critical line