↑

Properties

Label 8-288e4-1.1-c3e4-0-0
Degree 88
Conductor 68797071366879707136
Sign 11
Analytic cond. 83374.683374.6
Root an. cond. 4.122204.12220
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 40·7-s + 420·25-s − 248·31-s − 372·49-s + 120·73-s − 376·79-s + 520·97-s + 2.28e3·103-s + 2.44e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.74e3·169-s + 173-s − 1.68e4·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2.15·7-s + 3.35·25-s − 1.43·31-s − 1.08·49-s + 0.192·73-s − 0.535·79-s + 0.544·97-s + 2.18·103-s + 1.83·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.795·169-s + 0.000439·173-s − 7.25·175-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯

Functional equation

Λ(s)=((220⋅38)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((220⋅38)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅382^{20} \cdot 3^{8}
Sign: 11
Analytic conductor: 83374.683374.6
Root analytic conductor: 4.122204.12220
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅38, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{20} \cdot 3^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.29812137620.2981213762
L(12)L(\frac12) ≈\approx 0.29812137620.2981213762
L(52)L(\frac{5}{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
3 1 1
good5C22C_2^2 (1−42pT2+p6T4)2 ( 1 - 42 p T^{2} + p^{6} T^{4} )^{2}
7C2C_2 (1+10T+p3T2)4 ( 1 + 10 T + p^{3} T^{2} )^{4}
11C22C_2^2 (1−1222T2+p6T4)2 ( 1 - 1222 T^{2} + p^{6} T^{4} )^{2}
13C22C_2^2 (1−874T2+p6T4)2 ( 1 - 874 T^{2} + p^{6} T^{4} )^{2}
17C22C_2^2 (1+4194T2+p6T4)2 ( 1 + 4194 T^{2} + p^{6} T^{4} )^{2}
19C22C_2^2 (1+362T2+p6T4)2 ( 1 + 362 T^{2} + p^{6} T^{4} )^{2}
23C22C_2^2 (1+1806T2+p6T4)2 ( 1 + 1806 T^{2} + p^{6} T^{4} )^{2}
29C22C_2^2 (1+12062T2+p6T4)2 ( 1 + 12062 T^{2} + p^{6} T^{4} )^{2}
31C2C_2 (1+2pT+p3T2)4 ( 1 + 2 p T + p^{3} T^{2} )^{4}
37C22C_2^2 (1−97786T2+p6T4)2 ( 1 - 97786 T^{2} + p^{6} T^{4} )^{2}
41C22C_2^2 (1−2958T2+p6T4)2 ( 1 - 2958 T^{2} + p^{6} T^{4} )^{2}
43C22C_2^2 (1−144934T2+p6T4)2 ( 1 - 144934 T^{2} + p^{6} T^{4} )^{2}
47C22C_2^2 (1+4894T2+p6T4)2 ( 1 + 4894 T^{2} + p^{6} T^{4} )^{2}
53C22C_2^2 (1−280114T2+p6T4)2 ( 1 - 280114 T^{2} + p^{6} T^{4} )^{2}
59C22C_2^2 (1+127482T2+p6T4)2 ( 1 + 127482 T^{2} + p^{6} T^{4} )^{2}
61C22C_2^2 (1−168842T2+p6T4)2 ( 1 - 168842 T^{2} + p^{6} T^{4} )^{2}
67C22C_2^2 (1−94646T2+p6T4)2 ( 1 - 94646 T^{2} + p^{6} T^{4} )^{2}
71C2C_2 (1+p3T2)4 ( 1 + p^{3} T^{2} )^{4}
73C2C_2 (1−30T+p3T2)4 ( 1 - 30 T + p^{3} T^{2} )^{4}
79C2C_2 (1+94T+p3T2)4 ( 1 + 94 T + p^{3} T^{2} )^{4}
83C22C_2^2 (1−694134T2+p6T4)2 ( 1 - 694134 T^{2} + p^{6} T^{4} )^{2}
89C22C_2^2 (1+846738T2+p6T4)2 ( 1 + 846738 T^{2} + p^{6} T^{4} )^{2}
97C2C_2 (1−130T+p3T2)4 ( 1 - 130 T + p^{3} T^{2} )^{4}
show more
show less
   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.182955577806507176805367752384, −7.973105939005372437240550521558, −7.47797754408745442202415847960, −7.36837078979486428617197039598, −6.87359066905063971362898920330, −6.73590517830974794752686196014, −6.73018309164840051319366687763, −6.35275359434611735562757618653, −6.17949799467147215748418856336, −5.71787271307614744210991680586, −5.47208158849970056064865173978, −5.23983854672807806249488239578, −4.74174444524037927031743085510, −4.63733058652809752898203247451, −4.36404430933712176673413138925, −3.65774777900274509571381063927, −3.49911373794932381001709001421, −3.35143552432572228030071450784, −2.84771545662906970637974875852, −2.83972618018538893084396259883, −2.22932242136234250296507859662, −1.72726583032781484984080730914, −1.19651747452334224848339462325, −0.76095503809034163300793139936, −0.11598757161119102119435961137, 0.11598757161119102119435961137, 0.76095503809034163300793139936, 1.19651747452334224848339462325, 1.72726583032781484984080730914, 2.22932242136234250296507859662, 2.83972618018538893084396259883, 2.84771545662906970637974875852, 3.35143552432572228030071450784, 3.49911373794932381001709001421, 3.65774777900274509571381063927, 4.36404430933712176673413138925, 4.63733058652809752898203247451, 4.74174444524037927031743085510, 5.23983854672807806249488239578, 5.47208158849970056064865173978, 5.71787271307614744210991680586, 6.17949799467147215748418856336, 6.35275359434611735562757618653, 6.73018309164840051319366687763, 6.73590517830974794752686196014, 6.87359066905063971362898920330, 7.36837078979486428617197039598, 7.47797754408745442202415847960, 7.973105939005372437240550521558, 8.182955577806507176805367752384

Graph of the ZZ-function along the critical line