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Properties

Label 4-416e2-1.1-c1e2-0-8
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

−11.37168840793787288233172923128, −10.83228872062438302440557434694, −10.32442858055002501858250137540, −10.14138038224859624964234125400, −9.605097811540624682727093877165, −9.430494786937664535759407668439, −8.955636128101187260718184376317, −7.76497461112482200405099189371, −7.74632184836783213897782844514, −7.53689314953457584073246640950, −6.68219372843844639588494615195, −6.41491769231875701474624915748, −5.42281091134998164623675271986, −5.23526122000263984690340009282, −4.81797216331020672173860035645, −4.08821981060230659903242129669, −3.28019333367342851958767060337, −2.57871025879883010529356440435, −1.98700077270846129269061733720, −1.00743657042411183446450197924, 1.00743657042411183446450197924, 1.98700077270846129269061733720, 2.57871025879883010529356440435, 3.28019333367342851958767060337, 4.08821981060230659903242129669, 4.81797216331020672173860035645, 5.23526122000263984690340009282, 5.42281091134998164623675271986, 6.41491769231875701474624915748, 6.68219372843844639588494615195, 7.53689314953457584073246640950, 7.74632184836783213897782844514, 7.76497461112482200405099189371, 8.955636128101187260718184376317, 9.430494786937664535759407668439, 9.605097811540624682727093877165, 10.14138038224859624964234125400, 10.32442858055002501858250137540, 10.83228872062438302440557434694, 11.37168840793787288233172923128

Graph of the ZZ-function along the critical line