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Properties

Label 8-832e4-1.1-c2e4-0-1
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 264139.264139.
Root an. cond. 4.761334.76133
Motivic weight 22
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  − 8·5-s − 2·9-s + 84·17-s − 34·25-s + 48·29-s − 8·37-s − 32·41-s + 16·45-s − 34·49-s − 40·53-s − 64·61-s + 208·73-s + 49·81-s − 672·85-s − 352·89-s − 472·97-s + 80·101-s + 56·109-s + 56·113-s + 332·121-s + 528·125-s + 127-s + 131-s + 137-s + 139-s − 384·145-s + 149-s + ⋯
L(s)  = 1  − 8/5·5-s − 2/9·9-s + 4.94·17-s − 1.35·25-s + 1.65·29-s − 0.216·37-s − 0.780·41-s + 0.355·45-s − 0.693·49-s − 0.754·53-s − 1.04·61-s + 2.84·73-s + 0.604·81-s − 7.90·85-s − 3.95·89-s − 4.86·97-s + 0.792·101-s + 0.513·109-s + 0.495·113-s + 2.74·121-s + 4.22·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.64·145-s + 0.00671·149-s + ⋯

Functional equation

Λ(s)=((224⋅134)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((224⋅134)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 224⋅1342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 264139.264139.
Root analytic conductor: 4.761334.76133
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224⋅134, ( :1,1,1,1), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.7901473392.790147339
L(12)L(\frac12) ≈\approx 2.7901473392.790147339
L(2)L(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−pT2)2 ( 1 - p T^{2} )^{2}
good3D4D_{4}×\timesD4D_{4} (1−4T+p2T2−4p2T3+p4T4)(1+4T+p2T2+4p2T3+p4T4) ( 1 - 4 T + p^{2} T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )( 1 + 4 T + p^{2} T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )
5D4D_{4} (1+4T+41T2+4p2T3+p4T4)2 ( 1 + 4 T + 41 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
7C22≀C2C_2^2 \wr C_2 1+34T2+4259T4+34p4T6+p8T8 1 + 34 T^{2} + 4259 T^{4} + 34 p^{4} T^{6} + p^{8} T^{8}
11C22≀C2C_2^2 \wr C_2 1−332T2+53510T4−332p4T6+p8T8 1 - 332 T^{2} + 53510 T^{4} - 332 p^{4} T^{6} + p^{8} T^{8}
17C2C_2 (1−21T+p2T2)4 ( 1 - 21 T + p^{2} T^{2} )^{4}
19C22≀C2C_2^2 \wr C_2 1−364T2+24198T4−364p4T6+p8T8 1 - 364 T^{2} + 24198 T^{4} - 364 p^{4} T^{6} + p^{8} T^{8}
23C22≀C2C_2^2 \wr C_2 1−1268T2+821030T4−1268p4T6+p8T8 1 - 1268 T^{2} + 821030 T^{4} - 1268 p^{4} T^{6} + p^{8} T^{8}
29D4D_{4} (1−24T+1358T2−24p2T3+p4T4)2 ( 1 - 24 T + 1358 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2}
31C22≀C2C_2^2 \wr C_2 1−1412T2+1493510T4−1412p4T6+p8T8 1 - 1412 T^{2} + 1493510 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8}
37D4D_{4} (1+4T+1689T2+4p2T3+p4T4)2 ( 1 + 4 T + 1689 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
41D4D_{4} (1+16T+98T2+16p2T3+p4T4)2 ( 1 + 16 T + 98 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2}
43C22≀C2C_2^2 \wr C_2 1−3886T2+7765539T4−3886p4T6+p8T8 1 - 3886 T^{2} + 7765539 T^{4} - 3886 p^{4} T^{6} + p^{8} T^{8}
47C22≀C2C_2^2 \wr C_2 1+274T2+9637523T4+274p4T6+p8T8 1 + 274 T^{2} + 9637523 T^{4} + 274 p^{4} T^{6} + p^{8} T^{8}
53D4D_{4} (1+20T+4418T2+20p2T3+p4T4)2 ( 1 + 20 T + 4418 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
59C22≀C2C_2^2 \wr C_2 1−164pT2+43327878T4−164p5T6+p8T8 1 - 164 p T^{2} + 43327878 T^{4} - 164 p^{5} T^{6} + p^{8} T^{8}
61D4D_{4} (1+32T+6866T2+32p2T3+p4T4)2 ( 1 + 32 T + 6866 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2}
67C22≀C2C_2^2 \wr C_2 1−14572T2+92309766T4−14572p4T6+p8T8 1 - 14572 T^{2} + 92309766 T^{4} - 14572 p^{4} T^{6} + p^{8} T^{8}
71C22≀C2C_2^2 \wr C_2 1−15134T2+106881443T4−15134p4T6+p8T8 1 - 15134 T^{2} + 106881443 T^{4} - 15134 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1−104T+9150T2−104p2T3+p4T4)2 ( 1 - 104 T + 9150 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} )^{2}
79C22≀C2C_2^2 \wr C_2 1−5684T2+65469158T4−5684p4T6+p8T8 1 - 5684 T^{2} + 65469158 T^{4} - 5684 p^{4} T^{6} + p^{8} T^{8}
83C22≀C2C_2^2 \wr C_2 1−26260T2+267246150T4−26260p4T6+p8T8 1 - 26260 T^{2} + 267246150 T^{4} - 26260 p^{4} T^{6} + p^{8} T^{8}
89D4D_{4} (1+176T+21038T2+176p2T3+p4T4)2 ( 1 + 176 T + 21038 T^{2} + 176 p^{2} T^{3} + p^{4} T^{4} )^{2}
97D4D_{4} (1+236T+27542T2+236p2T3+p4T4)2 ( 1 + 236 T + 27542 T^{2} + 236 p^{2} T^{3} + p^{4} T^{4} )^{2}
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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

−7.15522794752902748767630561643, −7.14120948891458540015487316174, −6.54965721865656015388793012537, −6.53589119839647468450279366096, −6.17664655738859106786292590589, −5.87018462593299127664994200344, −5.69795919049884456658615334713, −5.49183175496949652932823065498, −5.31280678028012520490396732834, −5.04093999951866267038717424175, −4.80254673224194781424153517583, −4.39941217358321235486249047647, −4.15367380499473365964483031410, −3.80487398866009136006525505782, −3.79321338677471760427486714049, −3.42424844202454944310720482820, −3.14558843641988701394773690327, −2.96746127057233078744886705123, −2.86280551426022797477042193992, −2.21025785428274628577873873401, −1.73785585733562199840986072094, −1.34823875054861798318922661581, −1.24262007365230265857488816521, −0.57234663625823645418018803495, −0.38793990955894626315804131272, 0.38793990955894626315804131272, 0.57234663625823645418018803495, 1.24262007365230265857488816521, 1.34823875054861798318922661581, 1.73785585733562199840986072094, 2.21025785428274628577873873401, 2.86280551426022797477042193992, 2.96746127057233078744886705123, 3.14558843641988701394773690327, 3.42424844202454944310720482820, 3.79321338677471760427486714049, 3.80487398866009136006525505782, 4.15367380499473365964483031410, 4.39941217358321235486249047647, 4.80254673224194781424153517583, 5.04093999951866267038717424175, 5.31280678028012520490396732834, 5.49183175496949652932823065498, 5.69795919049884456658615334713, 5.87018462593299127664994200344, 6.17664655738859106786292590589, 6.53589119839647468450279366096, 6.54965721865656015388793012537, 7.14120948891458540015487316174, 7.15522794752902748767630561643

Graph of the ZZ-function along the critical line