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Properties

Label 8-425e4-1.1-c3e4-0-2
Degree 88
Conductor 3262539062532625390625
Sign 11
Analytic cond. 395384.395384.
Root an. cond. 5.007575.00757
Motivic weight 33
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 − 22·4-s + 140·8-s + 70·9-s − 14·13-s + 195·16-s + 35·17-s − 280·18-s + 14·19-s + 56·26-s − 2.77e3·32-s − 140·34-s − 1.54e3·36-s − 56·38-s + 592·43-s + 896·47-s + 341·49-s + 308·52-s + 630·53-s + 1.00e3·59-s + 1.04e3·64-s − 2.08e3·67-s − 770·68-s + 9.80e3·72-s − 308·76-s + 2.21e3·81-s − 2.14e3·83-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.75·4-s + 6.18·8-s + 2.59·9-s − 0.298·13-s + 3.04·16-s + 0.499·17-s − 3.66·18-s + 0.169·19-s + 0.422·26-s − 15.3·32-s − 0.706·34-s − 7.12·36-s − 0.239·38-s + 2.09·43-s + 2.78·47-s + 0.994·49-s + 0.821·52-s + 1.63·53-s + 2.22·59-s + 2.03·64-s − 3.79·67-s − 1.37·68-s + 16.0·72-s − 0.464·76-s + 3.04·81-s − 2.83·83-s + ⋯

Functional equation

Λ(s)=((58⋅174)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((58⋅174)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 58⋅1745^{8} \cdot 17^{4}
Sign: 11
Analytic conductor: 395384.395384.
Root analytic conductor: 5.007575.00757
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 58⋅174, ( :3/2,3/2,3/2,3/2), 1)(8,\ 5^{8} \cdot 17^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.53771574750.5377157475
L(12)L(\frac12) ≈\approx 0.53771574750.5377157475
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
17C22C_2^2 1−35T+280pT2−35p3T3+p6T4 1 - 35 T + 280 p T^{2} - 35 p^{3} T^{3} + p^{6} T^{4}
good2C2C_2 (1+T+p3T2)4 ( 1 + T + p^{3} T^{2} )^{4}
3C22C_2^2 (1−35T2+p6T4)2 ( 1 - 35 T^{2} + p^{6} T^{4} )^{2}
7D4×C2D_4\times C_2 1−341T2+1128T4−341p6T6+p12T8 1 - 341 T^{2} + 1128 T^{4} - 341 p^{6} T^{6} + p^{12} T^{8}
11D4×C2D_4\times C_2 1−4293T2+7887344T4−4293p6T6+p12T8 1 - 4293 T^{2} + 7887344 T^{4} - 4293 p^{6} T^{6} + p^{12} T^{8}
13D4D_{4} (1+7T−966T2+7p3T3+p6T4)2 ( 1 + 7 T - 966 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} )^{2}
19D4D_{4} (1−7T+8358T2−7p3T3+p6T4)2 ( 1 - 7 T + 8358 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} )^{2}
23D4×C2D_4\times C_2 1−5028T2+32833958T4−5028p6T6+p12T8 1 - 5028 T^{2} + 32833958 T^{4} - 5028 p^{6} T^{6} + p^{12} T^{8}
29C22C_2^2 (1−1058pT2+p6T4)2 ( 1 - 1058 p T^{2} + p^{6} T^{4} )^{2}
31D4×C2D_4\times C_2 1−32165T2+390771768T4−32165p6T6+p12T8 1 - 32165 T^{2} + 390771768 T^{4} - 32165 p^{6} T^{6} + p^{12} T^{8}
37D4×C2D_4\times C_2 1−91016T2+5345007678T4−91016p6T6+p12T8 1 - 91016 T^{2} + 5345007678 T^{4} - 91016 p^{6} T^{6} + p^{12} T^{8}
41D4×C2D_4\times C_2 1−169305T2+14486361728T4−169305p6T6+p12T8 1 - 169305 T^{2} + 14486361728 T^{4} - 169305 p^{6} T^{6} + p^{12} T^{8}
43C2C_2 (1−148T+p3T2)4 ( 1 - 148 T + p^{3} T^{2} )^{4}
47C2C_2 (1−224T+p3T2)4 ( 1 - 224 T + p^{3} T^{2} )^{4}
53D4D_{4} (1−315T+59320T2−315p3T3+p6T4)2 ( 1 - 315 T + 59320 T^{2} - 315 p^{3} T^{3} + p^{6} T^{4} )^{2}
59D4D_{4} (1−504T+130438T2−504p3T3+p6T4)2 ( 1 - 504 T + 130438 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} )^{2}
61D4×C2D_4\times C_2 1+91480T2+103275473118T4+91480p6T6+p12T8 1 + 91480 T^{2} + 103275473118 T^{4} + 91480 p^{6} T^{6} + p^{12} T^{8}
67D4D_{4} (1+1041T+609206T2+1041p3T3+p6T4)2 ( 1 + 1041 T + 609206 T^{2} + 1041 p^{3} T^{3} + p^{6} T^{4} )^{2}
71D4×C2D_4\times C_2 1−3103pT2+267650334444T4−3103p7T6+p12T8 1 - 3103 p T^{2} + 267650334444 T^{4} - 3103 p^{7} T^{6} + p^{12} T^{8}
73D4×C2D_4\times C_2 1−922873T2+419875572048T4−922873p6T6+p12T8 1 - 922873 T^{2} + 419875572048 T^{4} - 922873 p^{6} T^{6} + p^{12} T^{8}
79D4×C2D_4\times C_2 1−1820261T2+1309129438200T4−1820261p6T6+p12T8 1 - 1820261 T^{2} + 1309129438200 T^{4} - 1820261 p^{6} T^{6} + p^{12} T^{8}
83D4D_{4} (1+1071T+1424962T2+1071p3T3+p6T4)2 ( 1 + 1071 T + 1424962 T^{2} + 1071 p^{3} T^{3} + p^{6} T^{4} )^{2}
89D4D_{4} (1−931T+1363388T2−931p3T3+p6T4)2 ( 1 - 931 T + 1363388 T^{2} - 931 p^{3} T^{3} + p^{6} T^{4} )^{2}
97D4×C2D_4\times C_2 1−2397332T2+2743625522598T4−2397332p6T6+p12T8 1 - 2397332 T^{2} + 2743625522598 T^{4} - 2397332 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

−7.74281463466507980020438681541, −7.53280393488377820557947793233, −7.27131531805841697546062577595, −7.20791524297240793063666619916, −7.10542716812910151827950761916, −6.43592451714184319122213817662, −6.03733315647856846933151364470, −5.71993980255115565301646189438, −5.45206549910803858362535763924, −5.34127891848696507657038678410, −5.09232922850948785225852099744, −4.59725512627940550887162010613, −4.35360548567310295969794848037, −4.23904899194757325590043955363, −4.13974817229449330424588825595, −3.95794426081707943776524239940, −3.55511012498425988048020094737, −3.19401927856690496618639427293, −2.48305447237743809422276769437, −2.17728085808049780806942296527, −1.37019278287858476303060001564, −1.28827338895280955198509093465, −1.05088804339446745709676109229, −0.73024154617894875944534371062, −0.22749992464298863856864381722, 0.22749992464298863856864381722, 0.73024154617894875944534371062, 1.05088804339446745709676109229, 1.28827338895280955198509093465, 1.37019278287858476303060001564, 2.17728085808049780806942296527, 2.48305447237743809422276769437, 3.19401927856690496618639427293, 3.55511012498425988048020094737, 3.95794426081707943776524239940, 4.13974817229449330424588825595, 4.23904899194757325590043955363, 4.35360548567310295969794848037, 4.59725512627940550887162010613, 5.09232922850948785225852099744, 5.34127891848696507657038678410, 5.45206549910803858362535763924, 5.71993980255115565301646189438, 6.03733315647856846933151364470, 6.43592451714184319122213817662, 7.10542716812910151827950761916, 7.20791524297240793063666619916, 7.27131531805841697546062577595, 7.53280393488377820557947793233, 7.74281463466507980020438681541

Graph of the ZZ-function along the critical line