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Properties

Label 8-384e4-1.1-c4e4-0-0
Degree 88
Conductor 2174327193621743271936
Sign 11
Analytic cond. 2.48257×1062.48257\times 10^{6}
Root an. cond. 6.300326.30032
Motivic weight 44
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  + 12·3-s − 54·9-s − 536·11-s + 380·25-s − 2.05e3·27-s − 6.43e3·33-s − 6.72e3·49-s + 5.52e3·59-s + 1.95e4·73-s + 4.56e3·75-s − 8.82e3·81-s + 2.36e4·83-s + 2.34e4·97-s + 2.89e4·99-s − 8.50e4·107-s + 1.20e5·121-s + 127-s + 131-s + 137-s + 139-s − 8.06e4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.31e4·169-s + ⋯
L(s)  = 1  + 4/3·3-s − 2/3·9-s − 4.42·11-s + 0.607·25-s − 2.81·27-s − 5.90·33-s − 2.80·49-s + 1.58·59-s + 3.67·73-s + 0.810·75-s − 1.34·81-s + 3.43·83-s + 2.49·97-s + 2.95·99-s − 7.43·107-s + 8.26·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 3.73·147-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 3.26·169-s + ⋯

Functional equation

Λ(s)=((228⋅34)s/2 ΓC(s)4 L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}
Λ(s)=((228⋅34)s/2 ΓC(s+2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅342^{28} \cdot 3^{4}
Sign: 11
Analytic conductor: 2.48257×1062.48257\times 10^{6}
Root analytic conductor: 6.300326.30032
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅34, ( :2,2,2,2), 1)(8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.51893644090.5189364409
L(12)L(\frac12) ≈\approx 0.51893644090.5189364409
L(3)L(3) 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
3C2C_2 (1−2pT+p4T2)2 ( 1 - 2 p T + p^{4} T^{2} )^{2}
good5C22C_2^2 (1−38pT2+p8T4)2 ( 1 - 38 p T^{2} + p^{8} T^{4} )^{2}
7C22C_2^2 (1+3362T2+p8T4)2 ( 1 + 3362 T^{2} + p^{8} T^{4} )^{2}
11C2C_2 (1+134T+p4T2)4 ( 1 + 134 T + p^{4} T^{2} )^{4}
13C22C_2^2 (1+46558T2+p8T4)2 ( 1 + 46558 T^{2} + p^{8} T^{4} )^{2}
17C22C_2^2 (1−110594T2+p8T4)2 ( 1 - 110594 T^{2} + p^{8} T^{4} )^{2}
19C22C_2^2 (1−18434T2+p8T4)2 ( 1 - 18434 T^{2} + p^{8} T^{4} )^{2}
23C22C_2^2 (1−144962T2+p8T4)2 ( 1 - 144962 T^{2} + p^{8} T^{4} )^{2}
29C22C_2^2 (1+779522T2+p8T4)2 ( 1 + 779522 T^{2} + p^{8} T^{4} )^{2}
31C22C_2^2 (1+636002T2+p8T4)2 ( 1 + 636002 T^{2} + p^{8} T^{4} )^{2}
37C22C_2^2 (1−1156322T2+p8T4)2 ( 1 - 1156322 T^{2} + p^{8} T^{4} )^{2}
41C22C_2^2 (1−3988034T2+p8T4)2 ( 1 - 3988034 T^{2} + p^{8} T^{4} )^{2}
43C22C_2^2 (1−6657602T2+p8T4)2 ( 1 - 6657602 T^{2} + p^{8} T^{4} )^{2}
47C22C_2^2 (1−3123842T2+p8T4)2 ( 1 - 3123842 T^{2} + p^{8} T^{4} )^{2}
53C22C_2^2 (1+13118402T2+p8T4)2 ( 1 + 13118402 T^{2} + p^{8} T^{4} )^{2}
59C2C_2 (1−1382T+p4T2)4 ( 1 - 1382 T + p^{4} T^{2} )^{4}
61C22C_2^2 (1−27588002T2+p8T4)2 ( 1 - 27588002 T^{2} + p^{8} T^{4} )^{2}
67C22C_2^2 (1−40267394T2+p8T4)2 ( 1 - 40267394 T^{2} + p^{8} T^{4} )^{2}
71C22C_2^2 (1−47090882T2+p8T4)2 ( 1 - 47090882 T^{2} + p^{8} T^{4} )^{2}
73C2C_2 (1−4894T+p4T2)4 ( 1 - 4894 T + p^{4} T^{2} )^{4}
79C22C_2^2 (1+75709922T2+p8T4)2 ( 1 + 75709922 T^{2} + p^{8} T^{4} )^{2}
83C2C_2 (1−5914T+p4T2)4 ( 1 - 5914 T + p^{4} T^{2} )^{4}
89C2C_2 (1−146pT+p4T2)2(1+146pT+p4T2)2 ( 1 - 146 p T + p^{4} T^{2} )^{2}( 1 + 146 p T + p^{4} T^{2} )^{2}
97C2C_2 (1−5858T+p4T2)4 ( 1 - 5858 T + p^{4} T^{2} )^{4}
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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

−7.78332577230208700304679599811, −7.63010264034747106622935490636, −7.10430480391362138649301682272, −6.94236709201376229381263197796, −6.51188032419072245110733329907, −6.20249309461562686878591516958, −6.19745583611674279194586283094, −5.45946633727135401275480781455, −5.45125148743136476628050100445, −5.19985958850412922096076595441, −5.04777850826941420432333828960, −4.90323681184478135873501960048, −4.51948227567807503114102543737, −3.68890912417119097328602323510, −3.64851983149401973492872741171, −3.61305712144318260605117877105, −2.97404287040297724000134835028, −2.66718703119923065581349218555, −2.61632683857268691447442365977, −2.31973833912136690193194768659, −2.22808181711146725598583871078, −1.59186520771417424785865021970, −1.02888819555824534285800994232, −0.39269606082347173082454543518, −0.14272126131083889913452826852, 0.14272126131083889913452826852, 0.39269606082347173082454543518, 1.02888819555824534285800994232, 1.59186520771417424785865021970, 2.22808181711146725598583871078, 2.31973833912136690193194768659, 2.61632683857268691447442365977, 2.66718703119923065581349218555, 2.97404287040297724000134835028, 3.61305712144318260605117877105, 3.64851983149401973492872741171, 3.68890912417119097328602323510, 4.51948227567807503114102543737, 4.90323681184478135873501960048, 5.04777850826941420432333828960, 5.19985958850412922096076595441, 5.45125148743136476628050100445, 5.45946633727135401275480781455, 6.19745583611674279194586283094, 6.20249309461562686878591516958, 6.51188032419072245110733329907, 6.94236709201376229381263197796, 7.10430480391362138649301682272, 7.63010264034747106622935490636, 7.78332577230208700304679599811

Graph of the ZZ-function along the critical line