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Properties

Label 16-2640e8-1.1-c1e8-0-6
Degree 1616
Conductor 2.360×10272.360\times 10^{27}
Sign 11
Analytic cond. 3.89985×10103.89985\times 10^{10}
Root an. cond. 4.591354.59135
Motivic weight 11
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  + 8·5-s − 4·9-s + 36·25-s + 8·37-s − 32·45-s − 20·49-s + 40·53-s + 10·81-s − 16·89-s + 8·97-s + 88·113-s + 36·121-s + 120·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 68·169-s + 173-s + 179-s + 181-s + 64·185-s + ⋯
L(s)  = 1  + 3.57·5-s − 4/3·9-s + 36/5·25-s + 1.31·37-s − 4.77·45-s − 2.85·49-s + 5.49·53-s + 10/9·81-s − 1.69·89-s + 0.812·97-s + 8.27·113-s + 3.27·121-s + 10.7·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 4.70·185-s + ⋯

Functional equation

Λ(s)=((232⋅38⋅58⋅118)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅38⋅58⋅118)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 232⋅38⋅58⋅1182^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}
Sign: 11
Analytic conductor: 3.89985×10103.89985\times 10^{10}
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 232⋅38⋅58⋅118, ( :[1/2]8), 1)(16,\ 2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 20.9614331520.96143315
L(12)L(\frac12) ≈\approx 20.9614331520.96143315
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}
ppFp(T)F_p(T)
bad2 1 1
3 (1+T2)4 ( 1 + T^{2} )^{4}
5 (1−T)8 ( 1 - T )^{8}
11 (1−18T2+p2T4)2 ( 1 - 18 T^{2} + p^{2} T^{4} )^{2}
good7 (1+10T2+58T4+10p2T6+p4T8)2 ( 1 + 10 T^{2} + 58 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2}
13 (1−34T2+562T4−34p2T6+p4T8)2 ( 1 - 34 T^{2} + 562 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2}
17 (1−50T2+1138T4−50p2T6+p4T8)2 ( 1 - 50 T^{2} + 1138 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2}
19 (1+12T2+p2T4)4 ( 1 + 12 T^{2} + p^{2} T^{4} )^{4}
23 (1+40T2+1198T4+40p2T6+p4T8)2 ( 1 + 40 T^{2} + 1198 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2}
29 (1−48T2+p2T4)4 ( 1 - 48 T^{2} + p^{2} T^{4} )^{4}
31 (1−pT2)8 ( 1 - p T^{2} )^{8}
37 (1−2T+10T2−2pT3+p2T4)4 ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4}
41 (1−32T2−542T4−32p2T6+p4T8)2 ( 1 - 32 T^{2} - 542 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2}
43 (1+154T2+9562T4+154p2T6+p4T8)2 ( 1 + 154 T^{2} + 9562 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} )^{2}
47 (1−30T2+p2T4)4 ( 1 - 30 T^{2} + p^{2} T^{4} )^{4}
53 (1−10T+66T2−10pT3+p2T4)4 ( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{4}
59 (1−56T2+1246T4−56p2T6+p4T8)2 ( 1 - 56 T^{2} + 1246 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2}
61 (1−12T2−1882T4−12p2T6+p4T8)2 ( 1 - 12 T^{2} - 1882 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2}
67 (1−120T2+10238T4−120p2T6+p4T8)2 ( 1 - 120 T^{2} + 10238 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2}
71 (1−152T2+15598T4−152p2T6+p4T8)2 ( 1 - 152 T^{2} + 15598 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2}
73 (1−50T2−3342T4−50p2T6+p4T8)2 ( 1 - 50 T^{2} - 3342 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2}
79 (1+148T2+p2T4)4 ( 1 + 148 T^{2} + p^{2} T^{4} )^{4}
83 (1+274T2+31962T4+274p2T6+p4T8)2 ( 1 + 274 T^{2} + 31962 T^{4} + 274 p^{2} T^{6} + p^{4} T^{8} )^{2}
89 (1+2T+pT2)8 ( 1 + 2 T + p T^{2} )^{8}
97 (1−2T+130T2−2pT3+p2T4)4 ( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.60241940691337664756339118217, −3.57846592012896226064113728682, −3.38763652850889313629775705886, −3.19692656623676055493111689315, −3.14379964922152792643200760286, −3.02756629349794607707163487653, −3.01521114200887218548432875308, −2.81634523312050771679740027660, −2.60185968875867154312336421458, −2.57396483711030138302103505231, −2.41727817557103050911643753240, −2.21195431249332536310847064964, −2.13413428839076354756468100198, −2.08616407648655303914781932786, −2.07524902223853072074923319351, −1.88993409025940680727719968922, −1.76759714115748091924669606319, −1.42705030247846877707521834680, −1.33166229023194234486146826881, −1.17468005716145731613494875714, −1.05874062218772185016789856702, −0.836343177969994225917052606043, −0.74895398931156652786746570772, −0.40309120757923529974783926131, −0.27312590610215048942166256084, 0.27312590610215048942166256084, 0.40309120757923529974783926131, 0.74895398931156652786746570772, 0.836343177969994225917052606043, 1.05874062218772185016789856702, 1.17468005716145731613494875714, 1.33166229023194234486146826881, 1.42705030247846877707521834680, 1.76759714115748091924669606319, 1.88993409025940680727719968922, 2.07524902223853072074923319351, 2.08616407648655303914781932786, 2.13413428839076354756468100198, 2.21195431249332536310847064964, 2.41727817557103050911643753240, 2.57396483711030138302103505231, 2.60185968875867154312336421458, 2.81634523312050771679740027660, 3.01521114200887218548432875308, 3.02756629349794607707163487653, 3.14379964922152792643200760286, 3.19692656623676055493111689315, 3.38763652850889313629775705886, 3.57846592012896226064113728682, 3.60241940691337664756339118217

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.