↑

Properties

Label 8-832e4-1.1-c1e4-0-27
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 1948.051948.05
Root an. cond. 2.577502.57750
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  + 12·5-s − 2·9-s + 10·13-s − 6·17-s + 70·25-s + 6·29-s + 22·37-s − 18·41-s − 24·45-s − 10·49-s + 6·61-s + 120·65-s + 9·81-s − 72·85-s − 24·89-s + 48·97-s + 66·101-s − 40·109-s − 18·113-s − 20·117-s + 10·121-s + 240·125-s + 127-s + 131-s + 137-s + 139-s + 72·145-s + ⋯
L(s)  = 1  + 5.36·5-s − 2/3·9-s + 2.77·13-s − 1.45·17-s + 14·25-s + 1.11·29-s + 3.61·37-s − 2.81·41-s − 3.57·45-s − 1.42·49-s + 0.768·61-s + 14.8·65-s + 81-s − 7.80·85-s − 2.54·89-s + 4.87·97-s + 6.56·101-s − 3.83·109-s − 1.69·113-s − 1.84·117-s + 0.909·121-s + 21.4·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.97·145-s + ⋯

Functional equation

Λ(s)=((224⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 1948.051948.05
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 13.2803414813.28034148
L(12)L(\frac12) ≈\approx 13.2803414813.28034148
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 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
good3C23C_2^3 1+2T2−5T4+2p2T6+p4T8 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8}
5C2C_2 (1−3T+pT2)4 ( 1 - 3 T + p T^{2} )^{4}
7C23C_2^3 1+10T2+51T4+10p2T6+p4T8 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8}
11C23C_2^3 1−10T2−21T4−10p2T6+p4T8 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}
17C22C_2^2 (1+3T−8T2+3pT3+p2T4)2 ( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}
19C22C_2^2×\timesC22C_2^2 (1−37T2+p2T4)(1+11T2+p2T4) ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} )
23C23C_2^3 1+2T2−525T4+2p2T6+p4T8 1 + 2 T^{2} - 525 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8}
29C22C_2^2 (1−3T+32T2−3pT3+p2T4)2 ( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
31C22C_2^2 (1−46T2+p2T4)2 ( 1 - 46 T^{2} + p^{2} T^{4} )^{2}
37C2C_2 (1−10T+pT2)2(1−T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2}
41C22C_2^2 (1+9T+68T2+9pT3+p2T4)2 ( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}
43C22C_2^2×\timesC22C_2^2 (1−61T2+p2T4)(1+83T2+p2T4) ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} )
47C22C_2^2 (1−58T2+p2T4)2 ( 1 - 58 T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1+41T2+p2T4)2 ( 1 + 41 T^{2} + p^{2} T^{4} )^{2}
59C23C_2^3 1−106T2+7755T4−106p2T6+p4T8 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8}
61C22C_2^2 (1−3T+64T2−3pT3+p2T4)2 ( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}
67C23C_2^3 1−86T2+2907T4−86p2T6+p4T8 1 - 86 T^{2} + 2907 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8}
71C23C_2^3 1+106T2+6195T4+106p2T6+p4T8 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8}
73C22C_2^2 (1+T2+p2T4)2 ( 1 + T^{2} + p^{2} T^{4} )^{2}
79C22C_2^2 (1+146T2+p2T4)2 ( 1 + 146 T^{2} + p^{2} T^{4} )^{2}
83C22C_2^2 (1+118T2+p2T4)2 ( 1 + 118 T^{2} + p^{2} T^{4} )^{2}
89C22C_2^2 (1+12T+137T2+12pT3+p2T4)2 ( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}
97C2C_2 (1−19T+pT2)2(1−5T+pT2)2 ( 1 - 19 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2}
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

−7.04187050123495732505650361291, −6.89096421369472557559618420237, −6.50615787271399208738507079661, −6.48470779602025264408663185356, −6.36624380210407394217537853300, −6.03141076952646877595223168477, −5.94509160426653465885009543959, −5.91202368558215649598219805058, −5.83608773451308238364373758240, −5.15406000223628009449234090817, −5.10686273703375726937881321658, −4.93535309589584820946809917498, −4.78957187865614652179639619191, −4.22689262766705898670740590399, −3.93592406260603035914177269084, −3.52160472589566567952065896370, −3.33180932208966774494329845747, −2.83992475314975998045867389890, −2.55975754618398702944466960711, −2.36709004255334101646557394849, −2.17648547397391964082529955453, −1.76872828528182976199289387104, −1.53632524360471275931762260685, −1.29004654483349892036376943181, −0.854807221390718308225457398643, 0.854807221390718308225457398643, 1.29004654483349892036376943181, 1.53632524360471275931762260685, 1.76872828528182976199289387104, 2.17648547397391964082529955453, 2.36709004255334101646557394849, 2.55975754618398702944466960711, 2.83992475314975998045867389890, 3.33180932208966774494329845747, 3.52160472589566567952065896370, 3.93592406260603035914177269084, 4.22689262766705898670740590399, 4.78957187865614652179639619191, 4.93535309589584820946809917498, 5.10686273703375726937881321658, 5.15406000223628009449234090817, 5.83608773451308238364373758240, 5.91202368558215649598219805058, 5.94509160426653465885009543959, 6.03141076952646877595223168477, 6.36624380210407394217537853300, 6.48470779602025264408663185356, 6.50615787271399208738507079661, 6.89096421369472557559618420237, 7.04187050123495732505650361291

Graph of the ZZ-function along the critical line