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Properties

Label 4-3332e2-1.1-c1e2-0-2
Degree 44
Conductor 1110222411102224
Sign 11
Analytic cond. 707.887707.887
Root an. cond. 5.158115.15811
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s + 8·13-s − 6·17-s + 8·19-s + 2·25-s + 16·43-s + 24·47-s − 12·53-s − 8·67-s + 7·81-s − 24·89-s − 24·101-s − 16·103-s + 32·117-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 22·169-s + 32·171-s + ⋯
L(s)  = 1  + 4/3·9-s + 2.21·13-s − 1.45·17-s + 1.83·19-s + 2/5·25-s + 2.43·43-s + 3.50·47-s − 1.64·53-s − 0.977·67-s + 7/9·81-s − 2.54·89-s − 2.38·101-s − 1.57·103-s + 2.95·117-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 2.44·171-s + ⋯

Functional equation

Λ(s)=(11102224s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(11102224s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{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: 707.887707.887
Root analytic conductor: 5.158115.15811
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11102224, ( :1/2,1/2), 1)(4,\ 11102224,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.0739612754.073961275
L(12)L(\frac12) ≈\approx 4.0739612754.073961275
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
7 1 1
17C2C_2 1+6T+pT2 1 + 6 T + p T^{2}
good3C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
11C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C22C_2^2 1−44T2+p2T4 1 - 44 T^{2} + p^{2} T^{4}
29C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
31C22C_2^2 1−44T2+p2T4 1 - 44 T^{2} + p^{2} T^{4}
37C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
47C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C22C_2^2 1−92T2+p2T4 1 - 92 T^{2} + p^{2} T^{4}
73C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
79C22C_2^2 1−140T2+p2T4 1 - 140 T^{2} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
97C2C_2 (1−pT2)2 ( 1 - 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

−9.171924702527574337555717665426, −8.478853418615852924070730515768, −8.117150631464355883057345796486, −7.58244010693781146472357456071, −7.30460009731033303507926265645, −6.98330175918415174496554911145, −6.71585799606762800398881691780, −5.99073947578360479849892432441, −5.92416336827612259671591408493, −5.55971352873620900131879208340, −4.94533604186819072799102090721, −4.33556329033103752053931528617, −4.23651806592263843561796399330, −3.84723019786387171980301979499, −3.35557915310758406773997657842, −2.73350639811331207734201520511, −2.40401879973578705823224118485, −1.39188626362306616951778318912, −1.37450596698300350519268658808, −0.69197666760147965336248626507, 0.69197666760147965336248626507, 1.37450596698300350519268658808, 1.39188626362306616951778318912, 2.40401879973578705823224118485, 2.73350639811331207734201520511, 3.35557915310758406773997657842, 3.84723019786387171980301979499, 4.23651806592263843561796399330, 4.33556329033103752053931528617, 4.94533604186819072799102090721, 5.55971352873620900131879208340, 5.92416336827612259671591408493, 5.99073947578360479849892432441, 6.71585799606762800398881691780, 6.98330175918415174496554911145, 7.30460009731033303507926265645, 7.58244010693781146472357456071, 8.117150631464355883057345796486, 8.478853418615852924070730515768, 9.171924702527574337555717665426

Graph of the ZZ-function along the critical line