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Properties

Label 4-3332e2-1.1-c0e2-0-2
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 0.94432799540.9443279954
L(12)L(\frac12) ≈\approx 0.94432799540.9443279954
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

−9.293129452313635962850210808535, −8.936886535696815813808001273117, −8.119807390076456274170699016941, −7.79903300281211056905051076760, −7.62692154896981139874941901936, −7.19126732277324764819272267579, −6.61810494245726834085852894239, −6.48409094057237388477506045423, −5.73626471775706529259774505670, −5.53950092468412514274425129532, −5.12865094577231853148438181420, −5.01755391402236332033188650721, −4.42878025722632013149202268664, −4.24200820661517750504143262776, −3.26447926822368127229734667222, −3.00120623832427226630588368322, −2.38674720264578613763938481049, −1.97716610158020560763087206113, −1.77070181713590931447225921994, −0.55044963260828938184894677577, 0.55044963260828938184894677577, 1.77070181713590931447225921994, 1.97716610158020560763087206113, 2.38674720264578613763938481049, 3.00120623832427226630588368322, 3.26447926822368127229734667222, 4.24200820661517750504143262776, 4.42878025722632013149202268664, 5.01755391402236332033188650721, 5.12865094577231853148438181420, 5.53950092468412514274425129532, 5.73626471775706529259774505670, 6.48409094057237388477506045423, 6.61810494245726834085852894239, 7.19126732277324764819272267579, 7.62692154896981139874941901936, 7.79903300281211056905051076760, 8.119807390076456274170699016941, 8.936886535696815813808001273117, 9.293129452313635962850210808535

Graph of the ZZ-function along the critical line