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Properties

Label 4-162e2-1.1-c7e2-0-3
Degree 44
Conductor 2624426244
Sign 11
Analytic cond. 2561.002561.00
Root an. cond. 7.113817.11381
Motivic weight 77
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 − 165·5-s + 508·7-s − 512·8-s − 1.32e3·10-s + 3.02e3·11-s − 5.03e3·13-s + 4.06e3·14-s − 4.09e3·16-s + 6.37e3·17-s + 3.01e3·19-s + 2.41e4·22-s − 7.56e4·23-s + 7.81e4·25-s − 4.03e4·26-s − 8.26e4·29-s + 1.74e5·31-s + 5.10e4·34-s − 8.38e4·35-s − 6.47e5·37-s + 2.41e4·38-s + 8.44e4·40-s − 3.08e5·41-s − 3.36e5·43-s − 6.04e5·46-s − 3.83e5·47-s + 8.23e5·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.590·5-s + 0.559·7-s − 0.353·8-s − 0.417·10-s + 0.685·11-s − 0.636·13-s + 0.395·14-s − 1/4·16-s + 0.314·17-s + 0.100·19-s + 0.484·22-s − 1.29·23-s + 25-s − 0.449·26-s − 0.629·29-s + 1.05·31-s + 0.222·34-s − 0.330·35-s − 2.10·37-s + 0.0713·38-s + 0.208·40-s − 0.698·41-s − 0.645·43-s − 0.916·46-s − 0.538·47-s + 49-s + ⋯

Functional equation

Λ(s)=(26244s/2 ΓC(s)2 L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(26244s/2 ΓC(s+7/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2624426244    =    22⋅382^{2} \cdot 3^{8}
Sign: 11
Analytic conductor: 2561.002561.00
Root analytic conductor: 7.113817.11381
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 26244, ( :7/2,7/2), 1)(4,\ 26244,\ (\ :7/2, 7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 1.1958508491.195850849
L(12)L(\frac12) ≈\approx 1.1958508491.195850849
L(92)L(\frac{9}{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)
bad2C2C_2 1−p3T+p6T2 1 - p^{3} T + p^{6} T^{2}
3 1 1
good5C22C_2^2 1+33pT−2036p2T2+33p8T3+p14T4 1 + 33 p T - 2036 p^{2} T^{2} + 33 p^{8} T^{3} + p^{14} T^{4}
7C2C_2 (1−1763T+p7T2)(1+1255T+p7T2) ( 1 - 1763 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} )
11C22C_2^2 1−3024T−10342595T2−3024p7T3+p14T4 1 - 3024 T - 10342595 T^{2} - 3024 p^{7} T^{3} + p^{14} T^{4}
13C22C_2^2 1+5039T−37356996T2+5039p7T3+p14T4 1 + 5039 T - 37356996 T^{2} + 5039 p^{7} T^{3} + p^{14} T^{4}
17C2C_2 (1−3189T+p7T2)2 ( 1 - 3189 T + p^{7} T^{2} )^{2}
19C2C_2 (1−1508T+p7T2)2 ( 1 - 1508 T + p^{7} T^{2} )^{2}
23C22C_2^2 1+75600T+2310534553T2+75600p7T3+p14T4 1 + 75600 T + 2310534553 T^{2} + 75600 p^{7} T^{3} + p^{14} T^{4}
29C22C_2^2 1+82665T−10416374084T2+82665p7T3+p14T4 1 + 82665 T - 10416374084 T^{2} + 82665 p^{7} T^{3} + p^{14} T^{4}
31C22C_2^2 1−174892T+3074597553T2−174892p7T3+p14T4 1 - 174892 T + 3074597553 T^{2} - 174892 p^{7} T^{3} + p^{14} T^{4}
37C2C_2 (1+323569T+p7T2)2 ( 1 + 323569 T + p^{7} T^{2} )^{2}
41C22C_2^2 1+308118T−99817571957T2+308118p7T3+p14T4 1 + 308118 T - 99817571957 T^{2} + 308118 p^{7} T^{3} + p^{14} T^{4}
43C22C_2^2 1+336680T−158465188707T2+336680p7T3+p14T4 1 + 336680 T - 158465188707 T^{2} + 336680 p^{7} T^{3} + p^{14} T^{4}
47C22C_2^2 1+383196T−359783946047T2+383196p7T3+p14T4 1 + 383196 T - 359783946047 T^{2} + 383196 p^{7} T^{3} + p^{14} T^{4}
53C2C_2 (1+760206T+p7T2)2 ( 1 + 760206 T + p^{7} T^{2} )^{2}
59C22C_2^2 1+2225664T+2464928756077T2+2225664p7T3+p14T4 1 + 2225664 T + 2464928756077 T^{2} + 2225664 p^{7} T^{3} + p^{14} T^{4}
61C22C_2^2 1+2244815T+1896451548204T2+2244815p7T3+p14T4 1 + 2244815 T + 1896451548204 T^{2} + 2244815 p^{7} T^{3} + p^{14} T^{4}
67C22C_2^2 1+1473188T−3890428721979T2+1473188p7T3+p14T4 1 + 1473188 T - 3890428721979 T^{2} + 1473188 p^{7} T^{3} + p^{14} T^{4}
71C2C_2 (1−5006892T+p7T2)2 ( 1 - 5006892 T + p^{7} T^{2} )^{2}
73C2C_2 (1+5898301T+p7T2)2 ( 1 + 5898301 T + p^{7} T^{2} )^{2}
79C22C_2^2 1+7028768T+30199670611665T2+7028768p7T3+p14T4 1 + 7028768 T + 30199670611665 T^{2} + 7028768 p^{7} T^{3} + p^{14} T^{4}
83C22C_2^2 1+2651196T−20107210759211T2+2651196p7T3+p14T4 1 + 2651196 T - 20107210759211 T^{2} + 2651196 p^{7} T^{3} + p^{14} T^{4}
89C2C_2 (1−6770901T+p7T2)2 ( 1 - 6770901 T + p^{7} T^{2} )^{2}
97C22C_2^2 1+16176386T+180877179542883T2+16176386p7T3+p14T4 1 + 16176386 T + 180877179542883 T^{2} + 16176386 p^{7} T^{3} + p^{14} 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

−12.03719472501806546594764978369, −11.49773930708098193623026249306, −10.93820172749231544170830503072, −10.19578830033338212369629005631, −10.02156404243106517775191882813, −9.007174433624170436474232443470, −8.892118997898237631337935538337, −7.991495633654315643108948712125, −7.78593507555900304019491927372, −6.96327138131373984480491156613, −6.54307534890021345054734628436, −5.80945513924447681515046601577, −5.18812855186794106330884525122, −4.61741762966938221303184905023, −4.20613829575332397818940125706, −3.41020324991272017036920459122, −2.98527733613566380975822936027, −1.89341792425272311339430149905, −1.37365187520319000128052991652, −0.25687553343924753765171036965, 0.25687553343924753765171036965, 1.37365187520319000128052991652, 1.89341792425272311339430149905, 2.98527733613566380975822936027, 3.41020324991272017036920459122, 4.20613829575332397818940125706, 4.61741762966938221303184905023, 5.18812855186794106330884525122, 5.80945513924447681515046601577, 6.54307534890021345054734628436, 6.96327138131373984480491156613, 7.78593507555900304019491927372, 7.991495633654315643108948712125, 8.892118997898237631337935538337, 9.007174433624170436474232443470, 10.02156404243106517775191882813, 10.19578830033338212369629005631, 10.93820172749231544170830503072, 11.49773930708098193623026249306, 12.03719472501806546594764978369

Graph of the ZZ-function along the critical line