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Properties

Label 4-3240e2-1.1-c1e2-0-28
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 669.336669.336
Root an. cond. 5.086405.08640
Motivic weight 11
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  + 2·5-s + 2·7-s + 4·17-s − 4·19-s + 10·23-s + 3·25-s − 6·29-s + 12·31-s + 4·35-s − 12·37-s + 10·41-s + 4·43-s + 14·47-s − 5·49-s + 4·53-s + 12·59-s − 6·61-s − 2·67-s + 8·73-s − 4·79-s + 6·83-s + 8·85-s + 14·89-s − 8·95-s + 4·97-s + 4·101-s + 20·103-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s + 0.970·17-s − 0.917·19-s + 2.08·23-s + 3/5·25-s − 1.11·29-s + 2.15·31-s + 0.676·35-s − 1.97·37-s + 1.56·41-s + 0.609·43-s + 2.04·47-s − 5/7·49-s + 0.549·53-s + 1.56·59-s − 0.768·61-s − 0.244·67-s + 0.936·73-s − 0.450·79-s + 0.658·83-s + 0.867·85-s + 1.48·89-s − 0.820·95-s + 0.406·97-s + 0.398·101-s + 1.97·103-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 669.336669.336
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :1/2,1/2), 1)(4,\ 10497600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.7928905154.792890515
L(12)L(\frac12) ≈\approx 4.7928905154.792890515
L(32)L(\frac{3}{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)
bad2 1 1
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
good7D4D_{4} 1−2T+9T2−2pT3+p2T4 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19D4D_{4} 1+4T+18T2+4pT3+p2T4 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−10T+65T2−10pT3+p2T4 1 - 10 T + 65 T^{2} - 10 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+6T+43T2+6pT3+p2T4 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−12T+74T2−12pT3+p2T4 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41D4D_{4} 1−10T+83T2−10pT3+p2T4 1 - 10 T + 83 T^{2} - 10 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−4T−6T2−4pT3+p2T4 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−14T+137T2−14pT3+p2T4 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−4T+14T2−4pT3+p2T4 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−12T+130T2−12pT3+p2T4 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4}
61C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
67D4D_{4} 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
73D4D_{4} 1−8T+66T2−8pT3+p2T4 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+4T+138T2+4pT3+p2T4 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−6T+169T2−6pT3+p2T4 1 - 6 T + 169 T^{2} - 6 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−14T+131T2−14pT3+p2T4 1 - 14 T + 131 T^{2} - 14 p T^{3} + p^{2} T^{4}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
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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

−8.810094473244902225863958671273, −8.708854249634062873437283848740, −7.923685810313280565397389556111, −7.86872472112905429783454606288, −7.29561105727588080836615868616, −7.05642717890901162506008104396, −6.46727990629738871001075173803, −6.32629228830983142066046800463, −5.59886534851593283043038125112, −5.56554023586073573580900961129, −4.99341645588020936344215002534, −4.76715643472403162551356374357, −4.18775743972261135793208126128, −3.81827485177166097005396021959, −3.08777468354378049008845670357, −2.88796835386999305826815644374, −2.07683890516739483340949947699, −2.00506994364080859634970460265, −1.01004956177176863900695628427, −0.841177527231864577721067200852, 0.841177527231864577721067200852, 1.01004956177176863900695628427, 2.00506994364080859634970460265, 2.07683890516739483340949947699, 2.88796835386999305826815644374, 3.08777468354378049008845670357, 3.81827485177166097005396021959, 4.18775743972261135793208126128, 4.76715643472403162551356374357, 4.99341645588020936344215002534, 5.56554023586073573580900961129, 5.59886534851593283043038125112, 6.32629228830983142066046800463, 6.46727990629738871001075173803, 7.05642717890901162506008104396, 7.29561105727588080836615868616, 7.86872472112905429783454606288, 7.923685810313280565397389556111, 8.708854249634062873437283848740, 8.810094473244902225863958671273

Graph of the ZZ-function along the critical line