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Properties

Label 4-928e2-1.1-c0e2-0-0
Degree 44
Conductor 861184861184
Sign 11
Analytic cond. 0.2144910.214491
Root an. cond. 0.6805380.680538
Motivic weight 00
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  + 2·17-s + 2·25-s − 2·37-s + 2·41-s − 2·49-s − 2·61-s − 2·73-s − 81-s + 2·89-s − 2·97-s + 2·101-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·17-s + 2·25-s − 2·37-s + 2·41-s − 2·49-s − 2·61-s − 2·73-s − 81-s + 2·89-s − 2·97-s + 2·101-s − 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

Λ(s)=(861184s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 861184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(861184s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 861184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 861184861184    =    210⋅2922^{10} \cdot 29^{2}
Sign: 11
Analytic conductor: 0.2144910.214491
Root analytic conductor: 0.6805380.680538
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 861184, ( :0,0), 1)(4,\ 861184,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0320483491.032048349
L(12)L(\frac12) ≈\approx 1.0320483491.032048349
L(1)L(1) 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
29C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1+T4 1 + T^{4}
5C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
7C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
11C22C_2^2 1+T4 1 + T^{4}
13C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
17C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
19C22C_2^2 1+T4 1 + T^{4}
23C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
31C22C_2^2 1+T4 1 + T^{4}
37C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
41C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
43C22C_2^2 1+T4 1 + T^{4}
47C22C_2^2 1+T4 1 + T^{4}
53C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
59C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
61C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
67C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
79C22C_2^2 1+T4 1 + T^{4}
83C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
89C1C_1×\timesC2C_2 (1−T)2(1+T2) ( 1 - T )^{2}( 1 + T^{2} )
97C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{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

−10.36304149450379844748860503775, −10.32649361217922295847449140380, −9.468547787861388404157723393169, −9.373388442556614213884140929250, −8.800905635057467287936450614945, −8.466253626489076912498544923611, −7.78308213296023797723750067087, −7.70895793192274073880353785246, −7.08413554795742013658105486717, −6.74187740141601614423438985830, −6.06865082882127313601233434598, −5.83521575878393572733199904659, −5.07770353870020962471690943247, −4.96901346407292027055914794572, −4.26646666091638714178021999709, −3.63647565151377367456151427332, −3.01097054155806471359742676440, −2.87338733072793603172176381016, −1.71392705373248765773633153936, −1.15452819524465421065337968404, 1.15452819524465421065337968404, 1.71392705373248765773633153936, 2.87338733072793603172176381016, 3.01097054155806471359742676440, 3.63647565151377367456151427332, 4.26646666091638714178021999709, 4.96901346407292027055914794572, 5.07770353870020962471690943247, 5.83521575878393572733199904659, 6.06865082882127313601233434598, 6.74187740141601614423438985830, 7.08413554795742013658105486717, 7.70895793192274073880353785246, 7.78308213296023797723750067087, 8.466253626489076912498544923611, 8.800905635057467287936450614945, 9.373388442556614213884140929250, 9.468547787861388404157723393169, 10.32649361217922295847449140380, 10.36304149450379844748860503775

Graph of the ZZ-function along the critical line