↑

Properties

Label 8-867e4-1.1-c1e4-0-4
Degree 88
Conductor 565036352721565036352721
Sign 11
Analytic cond. 2297.122297.12
Root an. cond. 2.631162.63116
Motivic weight 11
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  + 6·4-s − 8·13-s + 19·16-s + 32·47-s − 48·52-s + 36·64-s + 48·67-s − 81-s + 40·89-s − 24·101-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 192·188-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 3·4-s − 2.21·13-s + 19/4·16-s + 4.66·47-s − 6.65·52-s + 9/2·64-s + 5.86·67-s − 1/9·81-s + 4.23·89-s − 2.38·101-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 − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 14.0·188-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

Λ(s)=((34⋅178)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅178)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅1783^{4} \cdot 17^{8}
Sign: 11
Analytic conductor: 2297.122297.12
Root analytic conductor: 2.631162.63116
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅178, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 17^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 7.0766954517.076695451
L(12)L(\frac12) ≈\approx 7.0766954517.076695451
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)
bad3C22C_2^2 1+T4 1 + T^{4}
17 1 1
good2C22C_2^2 (1−3T2+p2T4)2 ( 1 - 3 T^{2} + p^{2} T^{4} )^{2}
5C22C_2^2 (1+p2T4)2 ( 1 + p^{2} T^{4} )^{2}
7C23C_2^3 1−94T4+p4T8 1 - 94 T^{4} + p^{4} T^{8}
11C23C_2^3 1−206T4+p4T8 1 - 206 T^{4} + p^{4} T^{8}
13C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
19C22C_2^2 (1−22T2+p2T4)2 ( 1 - 22 T^{2} + p^{2} T^{4} )^{2}
23C23C_2^3 1−158T4+p4T8 1 - 158 T^{4} + p^{4} T^{8}
29C22C_2^2 (1+p2T4)2 ( 1 + p^{2} T^{4} )^{2}
31C23C_2^3 1+194T4+p4T8 1 + 194 T^{4} + p^{4} T^{8}
37C23C_2^3 1−2638T4+p4T8 1 - 2638 T^{4} + p^{4} T^{8}
41C22C_2^2×\timesC22C_2^2 (1−80T2+p2T4)(1+80T2+p2T4) ( 1 - 80 T^{2} + p^{2} T^{4} )( 1 + 80 T^{2} + p^{2} T^{4} )
43C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
47C2C_2 (1−8T+pT2)4 ( 1 - 8 T + p T^{2} )^{4}
53C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
59C22C_2^2 (1+26T2+p2T4)2 ( 1 + 26 T^{2} + p^{2} T^{4} )^{2}
61C23C_2^3 1−4078T4+p4T8 1 - 4078 T^{4} + p^{4} T^{8}
67C2C_2 (1−12T+pT2)4 ( 1 - 12 T + p T^{2} )^{4}
71C23C_2^3 1−10078T4+p4T8 1 - 10078 T^{4} + p^{4} T^{8}
73C22C_2^2 (1+p2T4)2 ( 1 + p^{2} T^{4} )^{2}
79C23C_2^3 1+7682T4+p4T8 1 + 7682 T^{4} + p^{4} T^{8}
83C22C_2^2 (1−22T2+p2T4)2 ( 1 - 22 T^{2} + p^{2} T^{4} )^{2}
89C2C_2 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}
97C23C_2^3 1−14974T4+p4T8 1 - 14974 T^{4} + p^{4} T^{8}
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

−7.24855481526897250176318709932, −7.09446953773647332066480738063, −6.82774272324287118620274547717, −6.65067942891167034305187767781, −6.49149435301475766187675589905, −6.33353520234343961750241681204, −5.94583115840277757570483074124, −5.71789386427427522169938518220, −5.41011017483418362932083092950, −5.40548779991955172021917267478, −5.02926382336883831668377067466, −4.74623200455836089600652149511, −4.54294338165024402937305069733, −3.99799860331990027746249921808, −3.77751178235424578980270167458, −3.65216790123326349616584524263, −3.31417160239880765780433016567, −2.75412167457608938169415456766, −2.50622774027584122798337491566, −2.40105704805085328042931097410, −2.36943668241269769636834931132, −2.06619823248521273630654774686, −1.53927791480247816916690213492, −1.04011391364982053581673139782, −0.60823112969553030483359783508, 0.60823112969553030483359783508, 1.04011391364982053581673139782, 1.53927791480247816916690213492, 2.06619823248521273630654774686, 2.36943668241269769636834931132, 2.40105704805085328042931097410, 2.50622774027584122798337491566, 2.75412167457608938169415456766, 3.31417160239880765780433016567, 3.65216790123326349616584524263, 3.77751178235424578980270167458, 3.99799860331990027746249921808, 4.54294338165024402937305069733, 4.74623200455836089600652149511, 5.02926382336883831668377067466, 5.40548779991955172021917267478, 5.41011017483418362932083092950, 5.71789386427427522169938518220, 5.94583115840277757570483074124, 6.33353520234343961750241681204, 6.49149435301475766187675589905, 6.65067942891167034305187767781, 6.82774272324287118620274547717, 7.09446953773647332066480738063, 7.24855481526897250176318709932

Graph of the ZZ-function along the critical line