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Properties

Label 8-3328e4-1.1-c1e4-0-3
Degree 88
Conductor 1.227×10141.227\times 10^{14}
Sign 11
Analytic cond. 498702.498702.
Root an. cond. 5.155015.15501
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  + 2·9-s − 12·17-s + 2·25-s − 18·49-s − 56·73-s − 15·81-s + 40·89-s − 8·97-s − 56·113-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.91·17-s + 2/5·25-s − 2.57·49-s − 6.55·73-s − 5/3·81-s + 4.23·89-s − 0.812·97-s − 5.26·113-s + 4/11·121-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 − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

Λ(s)=((232⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \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: 232⋅1342^{32} \cdot 13^{4}
Sign: 11
Analytic conductor: 498702.498702.
Root analytic conductor: 5.155015.15501
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 232⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{32} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.7557167411.755716741
L(12)L(\frac12) ≈\approx 1.7557167411.755716741
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+T2)2 ( 1 + T^{2} )^{2}
good3C22C_2^2 (1−T2+p2T4)2 ( 1 - T^{2} + p^{2} T^{4} )^{2}
5C22C_2^2 (1−T2+p2T4)2 ( 1 - T^{2} + p^{2} T^{4} )^{2}
7C22C_2^2 (1+9T2+p2T4)2 ( 1 + 9 T^{2} + p^{2} T^{4} )^{2}
11C22C_2^2 (1−2T2+p2T4)2 ( 1 - 2 T^{2} + p^{2} T^{4} )^{2}
17C2C_2 (1+3T+pT2)4 ( 1 + 3 T + p T^{2} )^{4}
19C22C_2^2 (1−18T2+p2T4)2 ( 1 - 18 T^{2} + p^{2} T^{4} )^{2}
23C22C_2^2 (1−34T2+p2T4)2 ( 1 - 34 T^{2} + p^{2} T^{4} )^{2}
29C2C_2 (1−4T+pT2)2(1+4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
31C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
37C22C_2^2 (1−65T2+p2T4)2 ( 1 - 65 T^{2} + p^{2} T^{4} )^{2}
41C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
43C22C_2^2 (1−41T2+p2T4)2 ( 1 - 41 T^{2} + p^{2} T^{4} )^{2}
47C22C_2^2 (1+89T2+p2T4)2 ( 1 + 89 T^{2} + p^{2} T^{4} )^{2}
53C2C_2 (1−14T+pT2)2(1+14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2}
59C22C_2^2 (1−98T2+p2T4)2 ( 1 - 98 T^{2} + p^{2} T^{4} )^{2}
61C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
67C22C_2^2 (1+46T2+p2T4)2 ( 1 + 46 T^{2} + p^{2} T^{4} )^{2}
71C22C_2^2 (1+97T2+p2T4)2 ( 1 + 97 T^{2} + p^{2} T^{4} )^{2}
73C2C_2 (1+14T+pT2)4 ( 1 + 14 T + p T^{2} )^{4}
79C22C_2^2 (1+78T2+p2T4)2 ( 1 + 78 T^{2} + p^{2} T^{4} )^{2}
83C22C_2^2 (1+154T2+p2T4)2 ( 1 + 154 T^{2} + p^{2} T^{4} )^{2}
89C2C_2 (1−10T+pT2)4 ( 1 - 10 T + p T^{2} )^{4}
97C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
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   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

−6.10746565822432930091051026768, −5.94253520243105416126131206293, −5.86379107399020928099026297478, −5.45387803173768776976184704064, −5.23116982894672747637051606040, −5.05303213584734202699689688518, −4.93670705825449601930891939898, −4.47870854996578677207664046055, −4.39125785254593678461574087875, −4.33468644254046648056491850628, −4.31850489629127107782029398352, −3.93073584593977016905232276533, −3.59811192331837045214888222113, −3.35588060359060981566985045467, −3.13000360775533887724957644628, −2.85817313257320118487583343621, −2.63571475388658294722741457533, −2.60814400776212100639895005951, −2.00428988594600100214551510481, −1.92462084059525295215857969172, −1.65138137018106984412920096306, −1.41432921684726186387629782177, −1.16725832193935022155067138059, −0.36781695494391838827899421765, −0.34352469053775353781739827941, 0.34352469053775353781739827941, 0.36781695494391838827899421765, 1.16725832193935022155067138059, 1.41432921684726186387629782177, 1.65138137018106984412920096306, 1.92462084059525295215857969172, 2.00428988594600100214551510481, 2.60814400776212100639895005951, 2.63571475388658294722741457533, 2.85817313257320118487583343621, 3.13000360775533887724957644628, 3.35588060359060981566985045467, 3.59811192331837045214888222113, 3.93073584593977016905232276533, 4.31850489629127107782029398352, 4.33468644254046648056491850628, 4.39125785254593678461574087875, 4.47870854996578677207664046055, 4.93670705825449601930891939898, 5.05303213584734202699689688518, 5.23116982894672747637051606040, 5.45387803173768776976184704064, 5.86379107399020928099026297478, 5.94253520243105416126131206293, 6.10746565822432930091051026768

Graph of the ZZ-function along the critical line