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Properties

Label 4-9576e2-1.1-c1e2-0-5
Degree 44
Conductor 9169977691699776
Sign 11
Analytic cond. 5846.855846.85
Root an. cond. 8.744418.74441
Motivic weight 11
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  − 2·5-s − 2·7-s + 2·17-s + 2·19-s + 8·23-s − 2·25-s + 10·29-s + 4·31-s + 4·35-s − 4·41-s + 16·43-s − 2·47-s + 3·49-s − 2·53-s − 8·59-s − 16·61-s − 12·67-s + 6·71-s − 24·79-s + 2·83-s − 4·85-s + 4·89-s − 4·95-s + 16·97-s + 14·101-s − 2·107-s + 12·109-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s + 0.485·17-s + 0.458·19-s + 1.66·23-s − 2/5·25-s + 1.85·29-s + 0.718·31-s + 0.676·35-s − 0.624·41-s + 2.43·43-s − 0.291·47-s + 3/7·49-s − 0.274·53-s − 1.04·59-s − 2.04·61-s − 1.46·67-s + 0.712·71-s − 2.70·79-s + 0.219·83-s − 0.433·85-s + 0.423·89-s − 0.410·95-s + 1.62·97-s + 1.39·101-s − 0.193·107-s + 1.14·109-s + ⋯

Functional equation

Λ(s)=(91699776s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(91699776s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9169977691699776    =    26⋅34⋅72⋅1922^{6} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 5846.855846.85
Root analytic conductor: 8.744418.74441
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 91699776, ( :1/2,1/2), 1)(4,\ 91699776,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6018595472.601859547
L(12)L(\frac12) ≈\approx 2.6018595472.601859547
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
7C1C_1 (1+T)2 ( 1 + T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1+2T+6T2+2pT3+p2T4 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
17D4D_{4} 1−2T+30T2−2pT3+p2T4 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−8T+42T2−8pT3+p2T4 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4}
29D4D_{4} 1−10T+78T2−10pT3+p2T4 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4}
31C4C_4 1−4T+46T2−4pT3+p2T4 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
41C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47D4D_{4} 1+2T+90T2+2pT3+p2T4 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+2T+102T2+2pT3+p2T4 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4}
59C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
61C4C_4 1+16T+166T2+16pT3+p2T4 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+12T+150T2+12pT3+p2T4 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−6T+146T2−6pT3+p2T4 1 - 6 T + 146 T^{2} - 6 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+126T2+p2T4 1 + 126 T^{2} + p^{2} T^{4}
79C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
83D4D_{4} 1−2T+122T2−2pT3+p2T4 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−4T+102T2−4pT3+p2T4 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−16T+238T2−16pT3+p2T4 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} 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

−7.70951484034051205536681935895, −7.52823803322846159759648154169, −7.18889788851027380560146848903, −7.02017564502477463725896315624, −6.34575278849340594344900133085, −6.29853771056009229805266866160, −5.80057983565297474381585157848, −5.63452705621850259837392143914, −4.92740482504015199487720013440, −4.68493701240582109385200006039, −4.41593323584033033673555554056, −4.13135328882348988959716002594, −3.41787713104582311019878868858, −3.17511968326727805533838633187, −3.01659255582841408667145002889, −2.62744838700480254328747733856, −1.91145173289312730176780547763, −1.40009131734888899170382690652, −0.74511465595496916241196092036, −0.52943922930130364121168689504, 0.52943922930130364121168689504, 0.74511465595496916241196092036, 1.40009131734888899170382690652, 1.91145173289312730176780547763, 2.62744838700480254328747733856, 3.01659255582841408667145002889, 3.17511968326727805533838633187, 3.41787713104582311019878868858, 4.13135328882348988959716002594, 4.41593323584033033673555554056, 4.68493701240582109385200006039, 4.92740482504015199487720013440, 5.63452705621850259837392143914, 5.80057983565297474381585157848, 6.29853771056009229805266866160, 6.34575278849340594344900133085, 7.02017564502477463725896315624, 7.18889788851027380560146848903, 7.52823803322846159759648154169, 7.70951484034051205536681935895

Graph of the ZZ-function along the critical line