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Properties

Label 8-275e4-1.1-c3e4-0-1
Degree 88
Conductor 57191406255719140625
Sign 11
Analytic cond. 69309.869309.8
Root an. cond. 4.028094.02809
Motivic weight 33
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  − 4-s + 95·9-s − 44·11-s + 81·16-s + 6·19-s + 442·29-s + 282·31-s − 95·36-s − 288·41-s + 44·44-s + 371·49-s + 172·59-s − 310·61-s − 225·64-s + 3.17e3·71-s − 6·76-s + 2.58e3·79-s + 5.34e3·81-s + 3.55e3·89-s − 4.18e3·99-s − 1.85e3·101-s + 1.64e3·109-s − 442·116-s + 1.21e3·121-s − 282·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/8·4-s + 3.51·9-s − 1.20·11-s + 1.26·16-s + 0.0724·19-s + 2.83·29-s + 1.63·31-s − 0.439·36-s − 1.09·41-s + 0.150·44-s + 1.08·49-s + 0.379·59-s − 0.650·61-s − 0.439·64-s + 5.30·71-s − 0.00905·76-s + 3.68·79-s + 7.33·81-s + 4.23·89-s − 4.24·99-s − 1.82·101-s + 1.44·109-s − 0.353·116-s + 0.909·121-s − 0.204·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=((58⋅114)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((58⋅114)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 58⋅1145^{8} \cdot 11^{4}
Sign: 11
Analytic conductor: 69309.869309.8
Root analytic conductor: 4.028094.02809
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 58⋅114, ( :3/2,3/2,3/2,3/2), 1)(8,\ 5^{8} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 10.2605165510.26051655
L(12)L(\frac12) ≈\approx 10.2605165510.26051655
L(52)L(\frac{5}{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)
bad5 1 1
11C1C_1 (1+pT)4 ( 1 + p T )^{4}
good2D4×C2D_4\times C_2 1+T2−5p4T4+p6T6+p12T8 1 + T^{2} - 5 p^{4} T^{4} + p^{6} T^{6} + p^{12} T^{8}
3D4×C2D_4\times C_2 1−95T2+3676T4−95p6T6+p12T8 1 - 95 T^{2} + 3676 T^{4} - 95 p^{6} T^{6} + p^{12} T^{8}
7D4×C2D_4\times C_2 1−53pT2+54552T4−53p7T6+p12T8 1 - 53 p T^{2} + 54552 T^{4} - 53 p^{7} T^{6} + p^{12} T^{8}
13D4×C2D_4\times C_2 1−6688T2+19773454T4−6688p6T6+p12T8 1 - 6688 T^{2} + 19773454 T^{4} - 6688 p^{6} T^{6} + p^{12} T^{8}
17D4×C2D_4\times C_2 1−5795T2+28665316T4−5795p6T6+p12T8 1 - 5795 T^{2} + 28665316 T^{4} - 5795 p^{6} T^{6} + p^{12} T^{8}
19D4D_{4} (1−3T+5114T2−3p3T3+p6T4)2 ( 1 - 3 T + 5114 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} )^{2}
23D4×C2D_4\times C_2 1−2060pT2+857131206T4−2060p7T6+p12T8 1 - 2060 p T^{2} + 857131206 T^{4} - 2060 p^{7} T^{6} + p^{12} T^{8}
29D4D_{4} (1−221T+39564T2−221p3T3+p6T4)2 ( 1 - 221 T + 39564 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} )^{2}
31D4D_{4} (1−141T+6374T2−141p3T3+p6T4)2 ( 1 - 141 T + 6374 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} )^{2}
37D4×C2D_4\times C_2 1−30655T2+2910831256T4−30655p6T6+p12T8 1 - 30655 T^{2} + 2910831256 T^{4} - 30655 p^{6} T^{6} + p^{12} T^{8}
41D4D_{4} (1+144T+39598T2+144p3T3+p6T4)2 ( 1 + 144 T + 39598 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} )^{2}
43D4×C2D_4\times C_2 1−109372T2+15260943894T4−109372p6T6+p12T8 1 - 109372 T^{2} + 15260943894 T^{4} - 109372 p^{6} T^{6} + p^{12} T^{8}
47D4×C2D_4\times C_2 1−329140T2+21975462p2T4−329140p6T6+p12T8 1 - 329140 T^{2} + 21975462 p^{2} T^{4} - 329140 p^{6} T^{6} + p^{12} T^{8}
53D4×C2D_4\times C_2 1−345895T2+72577931016T4−345895p6T6+p12T8 1 - 345895 T^{2} + 72577931016 T^{4} - 345895 p^{6} T^{6} + p^{12} T^{8}
59D4D_{4} (1−86T+306510T2−86p3T3+p6T4)2 ( 1 - 86 T + 306510 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} )^{2}
61D4D_{4} (1+155T+399784T2+155p3T3+p6T4)2 ( 1 + 155 T + 399784 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} )^{2}
67D4×C2D_4\times C_2 1−976424T2+412501678782T4−976424p6T6+p12T8 1 - 976424 T^{2} + 412501678782 T^{4} - 976424 p^{6} T^{6} + p^{12} T^{8}
71D4D_{4} (1−1587T+1329650T2−1587p3T3+p6T4)2 ( 1 - 1587 T + 1329650 T^{2} - 1587 p^{3} T^{3} + p^{6} T^{4} )^{2}
73D4×C2D_4\times C_2 1−1525024T2+884022947422T4−1525024p6T6+p12T8 1 - 1525024 T^{2} + 884022947422 T^{4} - 1525024 p^{6} T^{6} + p^{12} T^{8}
79D4D_{4} (1−1294T+1349454T2−1294p3T3+p6T4)2 ( 1 - 1294 T + 1349454 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4} )^{2}
83D4×C2D_4\times C_2 1−1511816T2+1128809299902T4−1511816p6T6+p12T8 1 - 1511816 T^{2} + 1128809299902 T^{4} - 1511816 p^{6} T^{6} + p^{12} T^{8}
89D4D_{4} (1−1777T+2191512T2−1777p3T3+p6T4)2 ( 1 - 1777 T + 2191512 T^{2} - 1777 p^{3} T^{3} + p^{6} T^{4} )^{2}
97D4×C2D_4\times C_2 1−1842928T2+2417317638046T4−1842928p6T6+p12T8 1 - 1842928 T^{2} + 2417317638046 T^{4} - 1842928 p^{6} T^{6} + p^{12} T^{8}
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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

−8.027931237161871424392257441560, −7.980775333456928087718060525912, −7.73216574883631815875090944214, −7.56588585431064936082739177274, −6.98980004613168497416407331010, −6.85581944849055318227905035804, −6.66123507156817981216777913513, −6.61284201340044846671770821204, −6.08810398602743430818688444520, −5.88275957735464605953974101622, −5.34444877455118688922572436657, −4.96016855131287772733825271484, −4.87008820175990593003947564022, −4.60183254571979644935984580149, −4.56920389883350203371250224072, −3.80884362878297945976619025775, −3.73225685174482483524338269278, −3.46382616359390149130769786299, −3.00890095887306138394614572558, −2.41681227933541063152020160258, −2.09739132238555092272592116348, −1.87125566795404075754337128300, −1.00029397773553886878634320796, −0.832830105290406522855055924617, −0.818520054576956611350462280492, 0.818520054576956611350462280492, 0.832830105290406522855055924617, 1.00029397773553886878634320796, 1.87125566795404075754337128300, 2.09739132238555092272592116348, 2.41681227933541063152020160258, 3.00890095887306138394614572558, 3.46382616359390149130769786299, 3.73225685174482483524338269278, 3.80884362878297945976619025775, 4.56920389883350203371250224072, 4.60183254571979644935984580149, 4.87008820175990593003947564022, 4.96016855131287772733825271484, 5.34444877455118688922572436657, 5.88275957735464605953974101622, 6.08810398602743430818688444520, 6.61284201340044846671770821204, 6.66123507156817981216777913513, 6.85581944849055318227905035804, 6.98980004613168497416407331010, 7.56588585431064936082739177274, 7.73216574883631815875090944214, 7.980775333456928087718060525912, 8.027931237161871424392257441560

Graph of the ZZ-function along the critical line