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Properties

Label 16-220e8-1.1-c1e8-0-3
Degree 1616
Conductor 5.488×10185.488\times 10^{18}
Sign 11
Analytic cond. 90.698190.6981
Root an. cond. 1.325401.32540
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  + 6·4-s − 8·5-s + 2·9-s + 19·16-s − 48·20-s + 36·25-s + 12·36-s + 4·37-s − 16·45-s − 34·49-s − 36·53-s + 36·64-s − 152·80-s + 19·81-s − 28·89-s + 16·97-s + 216·100-s + 80·113-s − 120·125-s + 127-s + 131-s + 137-s + 139-s + 38·144-s + 24·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 3·4-s − 3.57·5-s + 2/3·9-s + 19/4·16-s − 10.7·20-s + 36/5·25-s + 2·36-s + 0.657·37-s − 2.38·45-s − 4.85·49-s − 4.94·53-s + 9/2·64-s − 16.9·80-s + 19/9·81-s − 2.96·89-s + 1.62·97-s + 21.5·100-s + 7.52·113-s − 10.7·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 19/6·144-s + 1.97·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

Λ(s)=((216⋅58⋅118)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅58⋅118)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \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: 216⋅58⋅1182^{16} \cdot 5^{8} \cdot 11^{8}
Sign: 11
Analytic conductor: 90.698190.6981
Root analytic conductor: 1.325401.32540
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 216⋅58⋅118, ( :[1/2]8), 1)(16,\ 2^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.5737480541.573748054
L(12)L(\frac12) ≈\approx 1.5737480541.573748054
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 (1−3T2+p2T4)2 ( 1 - 3 T^{2} + p^{2} T^{4} )^{2}
5 (1+T)8 ( 1 + T )^{8}
11 1−178T4+p4T8 1 - 178 T^{4} + p^{4} T^{8}
good3 (1−T2−8T4−p2T6+p4T8)2 ( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2}
7 (1+17T2+144T4+17p2T6+p4T8)2 ( 1 + 17 T^{2} + 144 T^{4} + 17 p^{2} T^{6} + p^{4} T^{8} )^{2}
13 (1−22T2+p2T4)4 ( 1 - 22 T^{2} + p^{2} T^{4} )^{4}
17 (1−15T2+608T4−15p2T6+p4T8)2 ( 1 - 15 T^{2} + 608 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} )^{2}
19 (1+5T2+492T4+5p2T6+p4T8)2 ( 1 + 5 T^{2} + 492 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2}
23 (1−48T2+1214T4−48p2T6+p4T8)2 ( 1 - 48 T^{2} + 1214 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2}
29 (1−51T2+1676T4−51p2T6+p4T8)2 ( 1 - 51 T^{2} + 1676 T^{4} - 51 p^{2} T^{6} + p^{4} T^{8} )^{2}
31 (1−113T2+5088T4−113p2T6+p4T8)2 ( 1 - 113 T^{2} + 5088 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} )^{2}
37 (1−T+48T2−pT3+p2T4)4 ( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} )^{4}
41 (1−10T+pT2)4(1+10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4}
43 (1+128T2+7374T4+128p2T6+p4T8)2 ( 1 + 128 T^{2} + 7374 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2}
47 (1−144T2+9182T4−144p2T6+p4T8)2 ( 1 - 144 T^{2} + 9182 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2}
53 (1+9T+100T2+9pT3+p2T4)4 ( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4}
59 (1−80T2+4782T4−80p2T6+p4T8)2 ( 1 - 80 T^{2} + 4782 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2}
61 (1−107T2+5868T4−107p2T6+p4T8)2 ( 1 - 107 T^{2} + 5868 T^{4} - 107 p^{2} T^{6} + p^{4} T^{8} )^{2}
67 (1−74T2+p2T4)4 ( 1 - 74 T^{2} + p^{2} T^{4} )^{4}
71 (1−185T2+16512T4−185p2T6+p4T8)2 ( 1 - 185 T^{2} + 16512 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} )^{2}
73 (1−142T2+p2T4)4 ( 1 - 142 T^{2} + p^{2} T^{4} )^{4}
79 (1+140T2+10662T4+140p2T6+p4T8)2 ( 1 + 140 T^{2} + 10662 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2}
83 (1+106T2+p2T4)4 ( 1 + 106 T^{2} + p^{2} T^{4} )^{4}
89 (1+7T+164T2+7pT3+p2T4)4 ( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4}
97 (1−2T+pT2)8 ( 1 - 2 T + p T^{2} )^{8}
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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

−5.73812589338021216771892245360, −5.41627949932870405380881235136, −5.06489199016063625816894201352, −5.04850661913630374715683966257, −4.92241595564743753445556218881, −4.75649386291839883312593412153, −4.72864526896438551904392995787, −4.40838586209154441095090518196, −4.23848750455116035376804384910, −4.17876226032412850490795430184, −4.01466500307527323203232538551, −3.54531298097811069189650776905, −3.49951351285668897638779749280, −3.44574749391459005168799810958, −3.13452885624797282855874462960, −3.11865084233265932113536114836, −3.04524733287524019968809747837, −2.74383231450438970825911793480, −2.56109076747685554927755538732, −2.11771274823639781110416119215, −1.70978324994814382881067450747, −1.70850946274499909800647373114, −1.68299960504030342333200891621, −1.03754198720654088484727612719, −0.43139058921706822383338988331, 0.43139058921706822383338988331, 1.03754198720654088484727612719, 1.68299960504030342333200891621, 1.70850946274499909800647373114, 1.70978324994814382881067450747, 2.11771274823639781110416119215, 2.56109076747685554927755538732, 2.74383231450438970825911793480, 3.04524733287524019968809747837, 3.11865084233265932113536114836, 3.13452885624797282855874462960, 3.44574749391459005168799810958, 3.49951351285668897638779749280, 3.54531298097811069189650776905, 4.01466500307527323203232538551, 4.17876226032412850490795430184, 4.23848750455116035376804384910, 4.40838586209154441095090518196, 4.72864526896438551904392995787, 4.75649386291839883312593412153, 4.92241595564743753445556218881, 5.04850661913630374715683966257, 5.06489199016063625816894201352, 5.41627949932870405380881235136, 5.73812589338021216771892245360

Graph of the ZZ-function along the critical line

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