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Properties

Label 8-936e4-1.1-c1e4-0-19
Degree 88
Conductor 767544201216767544201216
Sign 11
Analytic cond. 3120.413120.41
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 12·7-s − 4·8-s + 24·14-s + 8·16-s − 8·17-s − 16·23-s − 4·25-s − 24·28-s − 20·31-s − 8·32-s + 16·34-s − 8·41-s + 32·46-s − 20·47-s + 68·49-s + 8·50-s + 48·56-s + 40·62-s + 8·64-s − 16·68-s + 12·71-s − 16·73-s + 8·79-s + 16·82-s − 32·92-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 4.53·7-s − 1.41·8-s + 6.41·14-s + 2·16-s − 1.94·17-s − 3.33·23-s − 4/5·25-s − 4.53·28-s − 3.59·31-s − 1.41·32-s + 2.74·34-s − 1.24·41-s + 4.71·46-s − 2.91·47-s + 68/7·49-s + 1.13·50-s + 6.41·56-s + 5.08·62-s + 64-s − 1.94·68-s + 1.42·71-s − 1.87·73-s + 0.900·79-s + 1.76·82-s − 3.33·92-s + ⋯

Functional equation

Λ(s)=((212⋅38⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅38⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \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: 212⋅38⋅1342^{12} \cdot 3^{8} \cdot 13^{4}
Sign: 11
Analytic conductor: 3120.413120.41
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 212⋅38⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{12} \cdot 3^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+pT+pT2+p2T3+p2T4 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}
3 1 1
13C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
good5C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
7D4D_{4} (1+6T+20T2+6pT3+p2T4)2 ( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}
11D4×C2D_4\times C_2 1−20T2+234T4−20p2T6+p4T8 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}
17D4D_{4} (1+4T+26T2+4pT3+p2T4)2 ( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
19D4×C2D_4\times C_2 1−68T2+1866T4−68p2T6+p4T8 1 - 68 T^{2} + 1866 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8}
23C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
29C22C_2^2 (1−54T2+p2T4)2 ( 1 - 54 T^{2} + p^{2} T^{4} )^{2}
31D4D_{4} (1+10T+84T2+10pT3+p2T4)2 ( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1−44T2+2454T4−44p2T6+p4T8 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8}
41D4D_{4} (1+4T+38T2+4pT3+p2T4)2 ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}
43D4×C2D_4\times C_2 1−116T2+6294T4−116p2T6+p4T8 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}
47D4D_{4} (1+10T+116T2+10pT3+p2T4)2 ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}
53D4×C2D_4\times C_2 1−84T2+4310T4−84p2T6+p4T8 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8}
59D4×C2D_4\times C_2 1−132T2+8618T4−132p2T6+p4T8 1 - 132 T^{2} + 8618 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8}
61D4×C2D_4\times C_2 1−116T2+7734T4−116p2T6+p4T8 1 - 116 T^{2} + 7734 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8}
67D4×C2D_4\times C_2 1−260T2+25866T4−260p2T6+p4T8 1 - 260 T^{2} + 25866 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8}
71D4D_{4} (1−6T+124T2−6pT3+p2T4)2 ( 1 - 6 T + 124 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}
73D4D_{4} (1+8T+150T2+8pT3+p2T4)2 ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
79D4D_{4} (1−4T+150T2−4pT3+p2T4)2 ( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}
83D4×C2D_4\times C_2 1−276T2+32522T4−276p2T6+p4T8 1 - 276 T^{2} + 32522 T^{4} - 276 p^{2} T^{6} + p^{4} T^{8}
89C22C_2^2 (1−122T2+p2T4)2 ( 1 - 122 T^{2} + p^{2} T^{4} )^{2}
97D4D_{4} (1+8T+102T2+8pT3+p2T4)2 ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} 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

−8.020941794939133584860518192864, −7.18706617342979307753758092314, −7.12450220630040367648499691281, −6.99854600532153647017091440480, −6.72006370074289614483089716787, −6.63784586481738135130986254858, −6.30362524727238835065348539218, −6.21000194626873928727063218920, −5.97233738629684496505553764677, −5.89668617770242784064811445644, −5.61637003307684074460220913949, −5.28293202589334371685282680310, −5.01435696178616976656579535507, −4.55837487039604919016566799370, −4.07346674060573431614367093117, −3.88022216881596183244909708408, −3.79463603611424123603995423590, −3.55815558896606981189675044694, −3.19448093856665652571622750036, −3.09355066450153156380876986634, −2.85657727637471904963457577851, −2.23791321779624822399883873520, −2.06517492412871521886346786147, −1.93801626642843690496073000937, −1.24190027439206878245471091921, 0, 0, 0, 0, 1.24190027439206878245471091921, 1.93801626642843690496073000937, 2.06517492412871521886346786147, 2.23791321779624822399883873520, 2.85657727637471904963457577851, 3.09355066450153156380876986634, 3.19448093856665652571622750036, 3.55815558896606981189675044694, 3.79463603611424123603995423590, 3.88022216881596183244909708408, 4.07346674060573431614367093117, 4.55837487039604919016566799370, 5.01435696178616976656579535507, 5.28293202589334371685282680310, 5.61637003307684074460220913949, 5.89668617770242784064811445644, 5.97233738629684496505553764677, 6.21000194626873928727063218920, 6.30362524727238835065348539218, 6.63784586481738135130986254858, 6.72006370074289614483089716787, 6.99854600532153647017091440480, 7.12450220630040367648499691281, 7.18706617342979307753758092314, 8.020941794939133584860518192864

Graph of the ZZ-function along the critical line