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Properties

Label 4-98e2-1.1-c1e2-0-2
Degree 44
Conductor 96049604
Sign 11
Analytic cond. 0.6123590.612359
Root an. cond. 0.8846090.884609
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-s + 2·3-s + 2·6-s − 8-s + 3·9-s − 8·13-s − 16-s − 6·17-s + 3·18-s − 2·19-s − 2·24-s + 5·25-s − 8·26-s + 10·27-s − 12·29-s + 4·31-s − 6·34-s − 2·37-s − 2·38-s − 16·39-s + 12·41-s + 16·43-s + 12·47-s − 2·48-s + 5·50-s − 12·51-s − 6·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.816·6-s − 0.353·8-s + 9-s − 2.21·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.458·19-s − 0.408·24-s + 25-s − 1.56·26-s + 1.92·27-s − 2.22·29-s + 0.718·31-s − 1.02·34-s − 0.328·37-s − 0.324·38-s − 2.56·39-s + 1.87·41-s + 2.43·43-s + 1.75·47-s − 0.288·48-s + 0.707·50-s − 1.68·51-s − 0.824·53-s + ⋯

Functional equation

Λ(s)=(9604s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9604s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 96049604    =    22⋅742^{2} \cdot 7^{4}
Sign: 11
Analytic conductor: 0.6123590.612359
Root analytic conductor: 0.8846090.884609
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 9604, ( :1/2,1/2), 1)(4,\ 9604,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6824496911.682449691
L(12)L(\frac12) ≈\approx 1.6824496911.682449691
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)
bad2C2C_2 1−T+T2 1 - T + T^{2}
7 1 1
good3C22C_2^2 1−2T+T2−2pT3+p2T4 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4}
5C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
17C22C_2^2 1+6T+19T2+6pT3+p2T4 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
29C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
31C2C_2 (1−11T+pT2)(1+7T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C22C_2^2 1+2T−33T2+2pT3+p2T4 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C22C_2^2 1−12T+97T2−12pT3+p2T4 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1−6T−23T2−6pT3+p2T4 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+8T+3T2+8pT3+p2T4 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−4T−51T2−4pT3+p2T4 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C22C_2^2 1+2T−69T2+2pT3+p2T4 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4}
79C22C_2^2 1+8T−15T2+8pT3+p2T4 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4}
83C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
89C22C_2^2 1−6T−53T2−6pT3+p2T4 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 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

−14.04145501722198037479383964324, −14.00599460194856240485032040018, −12.89897144252259646211093501170, −12.85516300122271479135410455197, −12.44533047287876342504227065817, −11.74368042716781372037397857686, −10.84462218398902176922880365263, −10.60537085923855477135106016776, −9.630199992252261854634945255997, −9.241789904092050155683227766971, −8.904163825273414140536527508881, −8.083333619699282901638240448876, −7.25391899305612172511439282965, −7.16393847545088947450776580615, −6.12510237641322476348159870666, −5.23488858771214743865463879013, −4.42448063842485448054280162476, −4.12098676425324769033179756617, −2.71904573650174489385286764559, −2.41749920092248519897743465838, 2.41749920092248519897743465838, 2.71904573650174489385286764559, 4.12098676425324769033179756617, 4.42448063842485448054280162476, 5.23488858771214743865463879013, 6.12510237641322476348159870666, 7.16393847545088947450776580615, 7.25391899305612172511439282965, 8.083333619699282901638240448876, 8.904163825273414140536527508881, 9.241789904092050155683227766971, 9.630199992252261854634945255997, 10.60537085923855477135106016776, 10.84462218398902176922880365263, 11.74368042716781372037397857686, 12.44533047287876342504227065817, 12.85516300122271479135410455197, 12.89897144252259646211093501170, 14.00599460194856240485032040018, 14.04145501722198037479383964324

Graph of the ZZ-function along the critical line