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Properties

Label 4-3344e2-1.1-c1e2-0-3
Degree 44
Conductor 1118233611182336
Sign 11
Analytic cond. 712.995712.995
Root an. cond. 5.167395.16739
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s − 2·9-s + 2·11-s − 7·13-s + 15-s + 2·17-s + 2·19-s + 21-s + 2·23-s − 6·25-s + 2·27-s − 15·29-s + 5·31-s − 2·33-s + 35-s + 16·37-s + 7·39-s − 11·41-s + 43-s + 2·45-s + 8·47-s − 10·49-s − 2·51-s + 6·53-s − 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.603·11-s − 1.94·13-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s + 0.417·23-s − 6/5·25-s + 0.384·27-s − 2.78·29-s + 0.898·31-s − 0.348·33-s + 0.169·35-s + 2.63·37-s + 1.12·39-s − 1.71·41-s + 0.152·43-s + 0.298·45-s + 1.16·47-s − 1.42·49-s − 0.280·51-s + 0.824·53-s − 0.269·55-s + ⋯

Functional equation

Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11182336s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1118233611182336    =    28⋅112⋅1922^{8} \cdot 11^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 712.995712.995
Root analytic conductor: 5.167395.16739
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 11182336, ( :1/2,1/2), 1)(4,\ 11182336,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
11C1C_1 (1−T)2 ( 1 - T )^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good3D4D_{4} 1+T+pT2+pT3+p2T4 1 + T + p T^{2} + p T^{3} + p^{2} T^{4}
5D4D_{4} 1+T+7T2+pT3+p2T4 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4}
7D4D_{4} 1+T+11T2+pT3+p2T4 1 + T + 11 T^{2} + p T^{3} + p^{2} T^{4}
13D4D_{4} 1+7T+35T2+7pT3+p2T4 1 + 7 T + 35 T^{2} + 7 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−2T+22T2−2pT3+p2T4 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−2T+34T2−2pT3+p2T4 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+15T+111T2+15pT3+p2T4 1 + 15 T + 111 T^{2} + 15 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−5T+39T2−5pT3+p2T4 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41D4D_{4} 1+11T+109T2+11pT3+p2T4 1 + 11 T + 109 T^{2} + 11 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−T+83T2−pT3+p2T4 1 - T + 83 T^{2} - p T^{3} + p^{2} T^{4}
47D4D_{4} 1−8T+58T2−8pT3+p2T4 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−6T+102T2−6pT3+p2T4 1 - 6 T + 102 T^{2} - 6 p T^{3} + p^{2} T^{4}
59C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67D4D_{4} 1+7T+65T2+7pT3+p2T4 1 + 7 T + 65 T^{2} + 7 p T^{3} + p^{2} T^{4}
71D4D_{4} 1−15T+169T2−15pT3+p2T4 1 - 15 T + 169 T^{2} - 15 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+26T+302T2+26pT3+p2T4 1 + 26 T + 302 T^{2} + 26 p T^{3} + p^{2} T^{4}
79D4D_{4} 1−2T+42T2−2pT3+p2T4 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+11T+167T2+11pT3+p2T4 1 + 11 T + 167 T^{2} + 11 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−6T+70T2−6pT3+p2T4 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4}
97C22C_2^2 1−14T2+p2T4 1 - 14 T^{2} + 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

−8.358895437090235206759436796860, −7.83200400972601025979297758630, −7.67019693490022117680034540987, −7.50131396194742971695020905811, −6.81706128950577403993615636276, −6.74562287064492887547690602924, −5.96821453965534371336900433131, −5.88375961149779338106889285144, −5.35753755528923034961581183438, −5.18609395933032407836473225252, −4.53935269492226908953650581988, −4.19418652498217995583052543980, −3.75051428546192640425754510789, −3.33235690774399847798854618134, −2.62687982149335497551463432781, −2.53341398825740151331511385314, −1.72584894135688654187460871474, −1.07731290027739869936185854387, 0, 0, 1.07731290027739869936185854387, 1.72584894135688654187460871474, 2.53341398825740151331511385314, 2.62687982149335497551463432781, 3.33235690774399847798854618134, 3.75051428546192640425754510789, 4.19418652498217995583052543980, 4.53935269492226908953650581988, 5.18609395933032407836473225252, 5.35753755528923034961581183438, 5.88375961149779338106889285144, 5.96821453965534371336900433131, 6.74562287064492887547690602924, 6.81706128950577403993615636276, 7.50131396194742971695020905811, 7.67019693490022117680034540987, 7.83200400972601025979297758630, 8.358895437090235206759436796860

Graph of the ZZ-function along the critical line