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Properties

Label 4-9702e2-1.1-c1e2-0-2
Degree 44
Conductor 9412880494128804
Sign 11
Analytic cond. 6001.736001.73
Root an. cond. 8.801758.80175
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  + 2·2-s + 3·4-s + 4·8-s − 2·11-s + 5·16-s − 4·22-s − 2·25-s − 16·29-s + 6·32-s + 12·37-s + 8·43-s − 6·44-s − 4·50-s + 12·53-s − 32·58-s + 7·64-s − 8·67-s + 12·71-s + 24·74-s + 24·79-s + 16·86-s − 8·88-s − 6·100-s + 24·106-s − 12·107-s − 32·109-s − 28·113-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 0.603·11-s + 5/4·16-s − 0.852·22-s − 2/5·25-s − 2.97·29-s + 1.06·32-s + 1.97·37-s + 1.21·43-s − 0.904·44-s − 0.565·50-s + 1.64·53-s − 4.20·58-s + 7/8·64-s − 0.977·67-s + 1.42·71-s + 2.78·74-s + 2.70·79-s + 1.72·86-s − 0.852·88-s − 3/5·100-s + 2.33·106-s − 1.16·107-s − 3.06·109-s − 2.63·113-s + ⋯

Functional equation

Λ(s)=(94128804s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(94128804s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9412880494128804    =    22⋅34⋅74⋅1122^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 6001.736001.73
Root analytic conductor: 8.801758.80175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 94128804, ( :1/2,1/2), 1)(4,\ 94128804,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.3081892646.308189264
L(12)L(\frac12) ≈\approx 6.3081892646.308189264
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)
bad2C1C_1 (1−T)2 ( 1 - T )^{2}
3 1 1
7 1 1
11C1C_1 (1+T)2 ( 1 + T )^{2}
good5C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
13C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
17C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
19C22C_2^2 1−12T2+p2T4 1 - 12 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
31C22C_2^2 1+60T2+p2T4 1 + 60 T^{2} + p^{2} T^{4}
37C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C22C_2^2 1+44T2+p2T4 1 + 44 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C22C_2^2 1+110T2+p2T4 1 + 110 T^{2} + p^{2} T^{4}
61C22C_2^2 1−40T2+p2T4 1 - 40 T^{2} + p^{2} T^{4}
67C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
79C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
83C22C_2^2 1+116T2+p2T4 1 + 116 T^{2} + p^{2} T^{4}
89C22C_2^2 1−64T2+p2T4 1 - 64 T^{2} + p^{2} T^{4}
97C22C_2^2 1+144T2+p2T4 1 + 144 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

−7.66581865878770086169520803952, −7.63362238073388942311955634834, −6.95409904947504520728113796786, −6.85653956103152510656696446210, −6.42388691104461960719565088928, −5.98614230469501452683895281319, −5.66037853163319986170706656983, −5.47713587041485881194559351213, −5.21461651676620261229250353823, −4.77714695405700510053804500914, −4.15724220995996450773411233157, −4.08799813158337190460095803665, −3.78623316392103469254275447438, −3.35297812324539984375493290858, −2.77528852017348765490566550807, −2.51814241954628856815631525932, −2.15208711338571205856404965829, −1.71261883885942142666576884402, −1.09588561555603425131919303949, −0.43077324012781284684716596598, 0.43077324012781284684716596598, 1.09588561555603425131919303949, 1.71261883885942142666576884402, 2.15208711338571205856404965829, 2.51814241954628856815631525932, 2.77528852017348765490566550807, 3.35297812324539984375493290858, 3.78623316392103469254275447438, 4.08799813158337190460095803665, 4.15724220995996450773411233157, 4.77714695405700510053804500914, 5.21461651676620261229250353823, 5.47713587041485881194559351213, 5.66037853163319986170706656983, 5.98614230469501452683895281319, 6.42388691104461960719565088928, 6.85653956103152510656696446210, 6.95409904947504520728113796786, 7.63362238073388942311955634834, 7.66581865878770086169520803952

Graph of the ZZ-function along the critical line