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Properties

Label 4-3332e2-1.1-c0e2-0-3
Degree 44
Conductor 1110222411102224
Sign 11
Analytic cond. 2.765182.76518
Root an. cond. 1.289521.28952
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  − 4-s − 2·5-s + 4·13-s + 16-s + 2·20-s + 2·25-s − 2·29-s + 2·37-s + 2·41-s − 4·52-s + 2·61-s − 64-s − 8·65-s − 2·73-s − 2·80-s − 81-s − 4·89-s − 2·97-s − 2·100-s + 4·101-s + 2·109-s − 2·113-s + 2·116-s − 2·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 4-s − 2·5-s + 4·13-s + 16-s + 2·20-s + 2·25-s − 2·29-s + 2·37-s + 2·41-s − 4·52-s + 2·61-s − 64-s − 8·65-s − 2·73-s − 2·80-s − 81-s − 4·89-s − 2·97-s − 2·100-s + 4·101-s + 2·109-s − 2·113-s + 2·116-s − 2·125-s + 127-s + 131-s + 137-s + ⋯

Functional equation

Λ(s)=(11102224s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(11102224s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1110222411102224    =    24⋅74⋅1722^{4} \cdot 7^{4} \cdot 17^{2}
Sign: 11
Analytic conductor: 2.765182.76518
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11102224, ( :0,0), 1)(4,\ 11102224,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.88561508760.8856150876
L(12)L(\frac12) ≈\approx 0.88561508760.8856150876
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)
bad2C2C_2 1+T2 1 + T^{2}
7 1 1
17C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1+T4 1 + T^{4}
5C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{2} )
11C22C_2^2 1+T4 1 + T^{4}
13C1C_1 (1−T)4 ( 1 - T )^{4}
19C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
23C22C_2^2 1+T4 1 + T^{4}
29C1C_1×\timesC2C_2 (1+T)2(1+T2) ( 1 + T )^{2}( 1 + T^{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} )
43C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
47C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
53C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{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}
71C22C_2^2 1+T4 1 + T^{4}
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 (1+T)4 ( 1 + T )^{4}
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

−8.790390094214946818624091041978, −8.548549548274563564144906219624, −8.274342717562745769936631149975, −8.098908181856115567616957731689, −7.48358014587181380502623646717, −7.41986238014984211437602090300, −6.84314331405163844154461594750, −6.15038956349376653971524218849, −6.00542313895195228226626409227, −5.69325487277978777709354685503, −5.26087062239580847145273039016, −4.40871171970737961750501582965, −4.13088693244731173451210324301, −3.94905446137667590140928639264, −3.85246361645720185831238690570, −3.11611928451943545657526856909, −2.96896215101482196034074936860, −1.77697439988707306132956663952, −1.15790871148051782974946018628, −0.71656054501944727655211663883, 0.71656054501944727655211663883, 1.15790871148051782974946018628, 1.77697439988707306132956663952, 2.96896215101482196034074936860, 3.11611928451943545657526856909, 3.85246361645720185831238690570, 3.94905446137667590140928639264, 4.13088693244731173451210324301, 4.40871171970737961750501582965, 5.26087062239580847145273039016, 5.69325487277978777709354685503, 6.00542313895195228226626409227, 6.15038956349376653971524218849, 6.84314331405163844154461594750, 7.41986238014984211437602090300, 7.48358014587181380502623646717, 8.098908181856115567616957731689, 8.274342717562745769936631149975, 8.548549548274563564144906219624, 8.790390094214946818624091041978

Graph of the ZZ-function along the critical line