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Properties

Label 8-39e8-1.1-c3e4-0-1
Degree 88
Conductor 5.352×10125.352\times 10^{12}
Sign 11
Analytic cond. 6.48606×1076.48606\times 10^{7}
Root an. cond. 9.473229.47322
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  − 3·4-s + 41·16-s + 216·17-s + 120·23-s − 228·25-s + 48·29-s − 1.06e3·43-s − 328·49-s + 864·53-s − 2.28e3·61-s − 411·64-s − 648·68-s + 288·79-s − 360·92-s + 684·100-s + 528·101-s + 3.24e3·107-s − 1.41e3·113-s − 144·116-s − 4.89e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 3/8·4-s + 0.640·16-s + 3.08·17-s + 1.08·23-s − 1.82·25-s + 0.307·29-s − 3.77·43-s − 0.956·49-s + 2.23·53-s − 4.78·61-s − 0.802·64-s − 1.15·68-s + 0.410·79-s − 0.407·92-s + 0.683·100-s + 0.520·101-s + 2.92·107-s − 1.17·113-s − 0.115·116-s − 3.67·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + ⋯

Functional equation

Λ(s)=((38⋅138)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((38⋅138)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 38⋅1383^{8} \cdot 13^{8}
Sign: 11
Analytic conductor: 6.48606×1076.48606\times 10^{7}
Root analytic conductor: 9.473229.47322
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 38⋅138, ( :3/2,3/2,3/2,3/2), 1)(8,\ 3^{8} \cdot 13^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.98642117080.9864211708
L(12)L(\frac12) ≈\approx 0.98642117080.9864211708
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)
bad3 1 1
13 1 1
good2D4×C2D_4\times C_2 1+3T2−p5T4+3p6T6+p12T8 1 + 3 T^{2} - p^{5} T^{4} + 3 p^{6} T^{6} + p^{12} T^{8}
5C22≀C2C_2^2 \wr C_2 1+228T2+33862T4+228p6T6+p12T8 1 + 228 T^{2} + 33862 T^{4} + 228 p^{6} T^{6} + p^{12} T^{8}
7C22≀C2C_2^2 \wr C_2 1+328T2+51918T4+328p6T6+p12T8 1 + 328 T^{2} + 51918 T^{4} + 328 p^{6} T^{6} + p^{12} T^{8}
11C22≀C2C_2^2 \wr C_2 1+4896T2+9533230T4+4896p6T6+p12T8 1 + 4896 T^{2} + 9533230 T^{4} + 4896 p^{6} T^{6} + p^{12} T^{8}
17C2C_2 (1−54T+p3T2)4 ( 1 - 54 T + p^{3} T^{2} )^{4}
19C22≀C2C_2^2 \wr C_2 1+15880T2+155242878T4+15880p6T6+p12T8 1 + 15880 T^{2} + 155242878 T^{4} + 15880 p^{6} T^{6} + p^{12} T^{8}
23D4D_{4} (1−60T+1870T2−60p3T3+p6T4)2 ( 1 - 60 T + 1870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2}
29D4D_{4} (1−24T+25558T2−24p3T3+p6T4)2 ( 1 - 24 T + 25558 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2}
31C22≀C2C_2^2 \wr C_2 1+102136T2+4357504590T4+102136p6T6+p12T8 1 + 102136 T^{2} + 4357504590 T^{4} + 102136 p^{6} T^{6} + p^{12} T^{8}
37C22≀C2C_2^2 \wr C_2 1+2164T2−2618990922T4+2164p6T6+p12T8 1 + 2164 T^{2} - 2618990922 T^{4} + 2164 p^{6} T^{6} + p^{12} T^{8}
41C22≀C2C_2^2 \wr C_2 1+230292T2+22372016182T4+230292p6T6+p12T8 1 + 230292 T^{2} + 22372016182 T^{4} + 230292 p^{6} T^{6} + p^{12} T^{8}
43D4D_{4} (1+532T+206406T2+532p3T3+p6T4)2 ( 1 + 532 T + 206406 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} )^{2}
47C22≀C2C_2^2 \wr C_2 1−85056T2+23167872958T4−85056p6T6+p12T8 1 - 85056 T^{2} + 23167872958 T^{4} - 85056 p^{6} T^{6} + p^{12} T^{8}
53D4D_{4} (1−432T+134134T2−432p3T3+p6T4)2 ( 1 - 432 T + 134134 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2}
59C22≀C2C_2^2 \wr C_2 1+716592T2+212420895502T4+716592p6T6+p12T8 1 + 716592 T^{2} + 212420895502 T^{4} + 716592 p^{6} T^{6} + p^{12} T^{8}
61D4D_{4} (1+1140T+12598pT2+1140p3T3+p6T4)2 ( 1 + 1140 T + 12598 p T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2}
67C22≀C2C_2^2 \wr C_2 1+475384T2+105182398878T4+475384p6T6+p12T8 1 + 475384 T^{2} + 105182398878 T^{4} + 475384 p^{6} T^{6} + p^{12} T^{8}
71C22≀C2C_2^2 \wr C_2 1+1283952T2+666179277502T4+1283952p6T6+p12T8 1 + 1283952 T^{2} + 666179277502 T^{4} + 1283952 p^{6} T^{6} + p^{12} T^{8}
73C22≀C2C_2^2 \wr C_2 1+762916T2+359882924838T4+762916p6T6+p12T8 1 + 762916 T^{2} + 359882924838 T^{4} + 762916 p^{6} T^{6} + p^{12} T^{8}
79D4D_{4} (1−144T+825118T2−144p3T3+p6T4)2 ( 1 - 144 T + 825118 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2}
83C22≀C2C_2^2 \wr C_2 1+1633872T2+191792062p2T4+1633872p6T6+p12T8 1 + 1633872 T^{2} + 191792062 p^{2} T^{4} + 1633872 p^{6} T^{6} + p^{12} T^{8}
89C22≀C2C_2^2 \wr C_2 1+2759412T2+2896886558902T4+2759412p6T6+p12T8 1 + 2759412 T^{2} + 2896886558902 T^{4} + 2759412 p^{6} T^{6} + p^{12} T^{8}
97C22≀C2C_2^2 \wr C_2 1−53564T2+1619815156998T4−53564p6T6+p12T8 1 - 53564 T^{2} + 1619815156998 T^{4} - 53564 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

−6.37930659496544574397705670879, −6.29721064039630347016816338798, −5.75838264734123836280183254451, −5.70776736150244280323402271041, −5.50181541502453252223599188487, −5.33535249208491691802886796430, −5.08633098353777881636386391772, −4.79053798433035189212045160954, −4.76605052051313146434766120682, −4.33345728255767385996132094379, −4.11462068124171021169062391522, −3.91861308170087015614107261167, −3.42519848662332914634031728735, −3.34162545487194498416355188288, −3.23048150081354990052309841992, −3.10532885501930015910677477839, −2.81039842989896806071099652258, −2.30695114666928373366558088683, −1.93389277118341985675288936185, −1.67920503356369532511487561673, −1.51764927910245479210637360112, −1.20828584253536194609831584715, −0.864712926976473255208328020719, −0.56706627024054596407793270340, −0.11453834143139711145686727803, 0.11453834143139711145686727803, 0.56706627024054596407793270340, 0.864712926976473255208328020719, 1.20828584253536194609831584715, 1.51764927910245479210637360112, 1.67920503356369532511487561673, 1.93389277118341985675288936185, 2.30695114666928373366558088683, 2.81039842989896806071099652258, 3.10532885501930015910677477839, 3.23048150081354990052309841992, 3.34162545487194498416355188288, 3.42519848662332914634031728735, 3.91861308170087015614107261167, 4.11462068124171021169062391522, 4.33345728255767385996132094379, 4.76605052051313146434766120682, 4.79053798433035189212045160954, 5.08633098353777881636386391772, 5.33535249208491691802886796430, 5.50181541502453252223599188487, 5.70776736150244280323402271041, 5.75838264734123836280183254451, 6.29721064039630347016816338798, 6.37930659496544574397705670879

Graph of the ZZ-function along the critical line