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Properties

Label 8-308e4-1.1-c1e4-0-1
Degree 88
Conductor 89991784968999178496
Sign 11
Analytic cond. 36.585636.5856
Root an. cond. 1.568241.56824
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2·4-s − 2·7-s − 5·8-s + 9-s + 4·11-s + 2·14-s + 5·16-s − 18-s − 4·22-s + 10·25-s − 4·28-s − 10·32-s + 2·36-s + 16·37-s + 8·43-s + 8·44-s − 11·49-s − 10·50-s + 8·53-s + 10·56-s − 2·63-s + 17·64-s − 5·72-s − 16·74-s − 8·77-s + 26·79-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s − 0.755·7-s − 1.76·8-s + 1/3·9-s + 1.20·11-s + 0.534·14-s + 5/4·16-s − 0.235·18-s − 0.852·22-s + 2·25-s − 0.755·28-s − 1.76·32-s + 1/3·36-s + 2.63·37-s + 1.21·43-s + 1.20·44-s − 1.57·49-s − 1.41·50-s + 1.09·53-s + 1.33·56-s − 0.251·63-s + 17/8·64-s − 0.589·72-s − 1.85·74-s − 0.911·77-s + 2.92·79-s + ⋯

Functional equation

Λ(s)=((28⋅74⋅114)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅74⋅114)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅74⋅1142^{8} \cdot 7^{4} \cdot 11^{4}
Sign: 11
Analytic conductor: 36.585636.5856
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅74⋅114, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{8} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.6317173141.631717314
L(12)L(\frac12) ≈\approx 1.6317173141.631717314
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)
bad2C22C_2^2 1+T−T2+pT3+p2T4 1 + T - T^{2} + p T^{3} + p^{2} T^{4}
7C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
11C22C_2^2 1−4T+5T2−4pT3+p2T4 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4}
good3C23C_2^3 1−T2−8T4−p2T6+p4T8 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8}
5C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
13C22C_2^2 (1−19T2+p2T4)2 ( 1 - 19 T^{2} + p^{2} T^{4} )^{2}
17C23C_2^3 1+6T2−253T4+6p2T6+p4T8 1 + 6 T^{2} - 253 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8}
19C22C_2^2 (1−pT2+p2T4)2 ( 1 - p T^{2} + p^{2} T^{4} )^{2}
23C22C_2^2×\timesC22C_2^2 (1−8T+41T2−8pT3+p2T4)(1+8T+41T2+8pT3+p2T4) ( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} )
29C22C_2^2 (1+5T2+p2T4)2 ( 1 + 5 T^{2} + p^{2} T^{4} )^{2}
31C23C_2^3 1−50T2+1539T4−50p2T6+p4T8 1 - 50 T^{2} + 1539 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8}
37C22C_2^2 (1−8T+27T2−8pT3+p2T4)2 ( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
41C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
43C2C_2 (1−2T+pT2)4 ( 1 - 2 T + p T^{2} )^{4}
47C22C_2^2 (1+pT2+p2T4)2 ( 1 + p T^{2} + p^{2} T^{4} )^{2}
53C22C_2^2 (1−4T−37T2−4pT3+p2T4)2 ( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
59C23C_2^3 1+111T2+8840T4+111p2T6+p4T8 1 + 111 T^{2} + 8840 T^{4} + 111 p^{2} T^{6} + p^{4} T^{8}
61C23C_2^3 1+115T2+9504T4+115p2T6+p4T8 1 + 115 T^{2} + 9504 T^{4} + 115 p^{2} T^{6} + p^{4} T^{8}
67C23C_2^3 1+127T2+11640T4+127p2T6+p4T8 1 + 127 T^{2} + 11640 T^{4} + 127 p^{2} T^{6} + p^{4} T^{8}
71C2C_2 (1−16T+pT2)2(1+16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2}
73C23C_2^3 1+34T2−4173T4+34p2T6+p4T8 1 + 34 T^{2} - 4173 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8}
79C2C_2 (1−17T+pT2)2(1+4T+pT2)2 ( 1 - 17 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
83C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
89C22C_2^2 (1−14T+107T2−14pT3+p2T4)2 ( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}
97C2C_2 (1−7T+pT2)4 ( 1 - 7 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

−8.688106580529564596149193465987, −8.254000898637319667647642313932, −7.939997294723289162588479863487, −7.69520832298764954453756809465, −7.69292817981257631736683171156, −7.04694028035111952244533651038, −7.02205138122538449900217255525, −6.55452601601042274363952623370, −6.54056845175998895519120571389, −6.24119971776377700456157998315, −6.17491161113186027297624416531, −5.77926771877885086278316910738, −5.28096265557092402268158243804, −5.15927870358845290077239650599, −4.73289205300307711125971517374, −4.30139628312825422825387016938, −4.10499508352843310201253655710, −3.63015172122245793587229739462, −3.23869005904284853338157907331, −3.16676602594659353979993680610, −2.68344781015173082511399528384, −2.16637211343685779202692696727, −2.05172014348148192776193157999, −0.985205700718565601513783445839, −0.855701828923577491365219203156, 0.855701828923577491365219203156, 0.985205700718565601513783445839, 2.05172014348148192776193157999, 2.16637211343685779202692696727, 2.68344781015173082511399528384, 3.16676602594659353979993680610, 3.23869005904284853338157907331, 3.63015172122245793587229739462, 4.10499508352843310201253655710, 4.30139628312825422825387016938, 4.73289205300307711125971517374, 5.15927870358845290077239650599, 5.28096265557092402268158243804, 5.77926771877885086278316910738, 6.17491161113186027297624416531, 6.24119971776377700456157998315, 6.54056845175998895519120571389, 6.55452601601042274363952623370, 7.02205138122538449900217255525, 7.04694028035111952244533651038, 7.69292817981257631736683171156, 7.69520832298764954453756809465, 7.939997294723289162588479863487, 8.254000898637319667647642313932, 8.688106580529564596149193465987

Graph of the ZZ-function along the critical line