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Properties

Label 4-416e2-1.1-c1e2-0-14
Degree 44
Conductor 173056173056
Sign 11
Analytic cond. 11.034211.0342
Root an. cond. 1.822571.82257
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  − 2·5-s + 5·9-s + 4·11-s + 6·13-s + 6·17-s + 12·23-s − 7·25-s − 6·37-s − 10·45-s + 5·49-s − 8·55-s + 20·59-s − 12·65-s − 24·67-s + 16·81-s − 32·83-s − 12·85-s + 20·99-s − 8·103-s − 30·109-s − 12·113-s − 24·115-s + 30·117-s − 10·121-s + 26·125-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.894·5-s + 5/3·9-s + 1.20·11-s + 1.66·13-s + 1.45·17-s + 2.50·23-s − 7/5·25-s − 0.986·37-s − 1.49·45-s + 5/7·49-s − 1.07·55-s + 2.60·59-s − 1.48·65-s − 2.93·67-s + 16/9·81-s − 3.51·83-s − 1.30·85-s + 2.01·99-s − 0.788·103-s − 2.87·109-s − 1.12·113-s − 2.23·115-s + 2.77·117-s − 0.909·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(173056s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(173056s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 173056173056    =    210⋅1322^{10} \cdot 13^{2}
Sign: 11
Analytic conductor: 11.034211.0342
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 173056, ( :1/2,1/2), 1)(4,\ 173056,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0875691942.087569194
L(12)L(\frac12) ≈\approx 2.0875691942.087569194
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
13C2C_2 1−6T+pT2 1 - 6 T + p T^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
5C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
7C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
37C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
41C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
43C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
47C22C_2^2 1−45T2+p2T4 1 - 45 T^{2} + p^{2} T^{4}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
71C22C_2^2 1−117T2+p2T4 1 - 117 T^{2} + p^{2} T^{4}
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C2C_2 (1+16T+pT2)2 ( 1 + 16 T + p T^{2} )^{2}
89C22C_2^2 1−162T2+p2T4 1 - 162 T^{2} + p^{2} T^{4}
97C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p 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

−11.51934658035853730324972508152, −11.02002109623427657805421566937, −10.45332787133852351827811142108, −10.25192866628245335004599790864, −9.499410376287534354260725032362, −9.279794348719621932061108838309, −8.618797312506838930781956045142, −8.342221936283905436847609548821, −7.55506918318717107045340125087, −7.34864537020561679716376479234, −6.80857743730259293062239842643, −6.45057159331266784874117649187, −5.60103366058374838686862810103, −5.28237839942400213115685528030, −4.22324818397657834157527136281, −4.06480225066723033621650033340, −3.61102415332154168227078522061, −2.92416774946599500699904970660, −1.35309831046559301069422652713, −1.29470269968441608072715738381, 1.29470269968441608072715738381, 1.35309831046559301069422652713, 2.92416774946599500699904970660, 3.61102415332154168227078522061, 4.06480225066723033621650033340, 4.22324818397657834157527136281, 5.28237839942400213115685528030, 5.60103366058374838686862810103, 6.45057159331266784874117649187, 6.80857743730259293062239842643, 7.34864537020561679716376479234, 7.55506918318717107045340125087, 8.342221936283905436847609548821, 8.618797312506838930781956045142, 9.279794348719621932061108838309, 9.499410376287534354260725032362, 10.25192866628245335004599790864, 10.45332787133852351827811142108, 11.02002109623427657805421566937, 11.51934658035853730324972508152

Graph of the ZZ-function along the critical line