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Properties

Label 8-416e4-1.1-c1e4-0-4
Degree 88
Conductor 2994837913629948379136
Sign 11
Analytic cond. 121.753121.753
Root an. cond. 1.822571.82257
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  − 2·3-s + 6·7-s + 5·9-s − 2·11-s − 4·13-s + 6·17-s + 6·19-s − 12·21-s − 6·23-s − 4·25-s − 10·27-s − 6·29-s + 4·33-s − 2·37-s + 8·39-s + 6·41-s − 6·43-s − 24·47-s + 21·49-s − 12·51-s − 12·57-s + 6·59-s − 14·61-s + 30·63-s + 18·67-s + 12·69-s − 6·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.26·7-s + 5/3·9-s − 0.603·11-s − 1.10·13-s + 1.45·17-s + 1.37·19-s − 2.61·21-s − 1.25·23-s − 4/5·25-s − 1.92·27-s − 1.11·29-s + 0.696·33-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 0.914·43-s − 3.50·47-s + 3·49-s − 1.68·51-s − 1.58·57-s + 0.781·59-s − 1.79·61-s + 3.77·63-s + 2.19·67-s + 1.44·69-s − 0.712·71-s + ⋯

Functional equation

Λ(s)=((220⋅134)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((220⋅134)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \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: 220⋅1342^{20} \cdot 13^{4}
Sign: 11
Analytic conductor: 121.753121.753
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅134, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{20} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.8364941681.836494168
L(12)L(\frac12) ≈\approx 1.8364941681.836494168
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+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
good3D4×C2D_4\times C_2 1+2T−T2−2T3+4T4−2pT5−p2T6+2p3T7+p4T8 1 + 2 T - T^{2} - 2 T^{3} + 4 T^{4} - 2 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
5C22C_2^2 (1+2T2+p2T4)2 ( 1 + 2 T^{2} + p^{2} T^{4} )^{2}
7D4×C2D_4\times C_2 1−6T+15T2−6pT3+20pT4−6p2T5+15p2T6−6p3T7+p4T8 1 - 6 T + 15 T^{2} - 6 p T^{3} + 20 p T^{4} - 6 p^{2} T^{5} + 15 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
11D4×C2D_4\times C_2 1+2T−T2−34T3−140T4−34pT5−p2T6+2p3T7+p4T8 1 + 2 T - T^{2} - 34 T^{3} - 140 T^{4} - 34 p T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
17D4×C2D_4\times C_2 1−6T+T2−6T3+324T4−6pT5+p2T6−6p3T7+p4T8 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 6 p T^{5} + p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
19D4×C2D_4\times C_2 1−6T+7T2+54T3−204T4+54pT5+7p2T6−6p3T7+p4T8 1 - 6 T + 7 T^{2} + 54 T^{3} - 204 T^{4} + 54 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
23D4×C2D_4\times C_2 1+6T−T2−54T3+12T4−54pT5−p2T6+6p3T7+p4T8 1 + 6 T - T^{2} - 54 T^{3} + 12 T^{4} - 54 p T^{5} - p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
29D4×C2D_4\times C_2 1+6T+T2−138T3−660T4−138pT5+p2T6+6p3T7+p4T8 1 + 6 T + T^{2} - 138 T^{3} - 660 T^{4} - 138 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
31C22C_2^2 (1+30T2+p2T4)2 ( 1 + 30 T^{2} + p^{2} T^{4} )^{2}
37D4×C2D_4\times C_2 1+2T+T2−142T3−1508T4−142pT5+p2T6+2p3T7+p4T8 1 + 2 T + T^{2} - 142 T^{3} - 1508 T^{4} - 142 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
41D4×C2D_4\times C_2 1−6T−47T2−6T3+3732T4−6pT5−47p2T6−6p3T7+p4T8 1 - 6 T - 47 T^{2} - 6 T^{3} + 3732 T^{4} - 6 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
43D4×C2D_4\times C_2 1+6T−9T2−246T3−1372T4−246pT5−9p2T6+6p3T7+p4T8 1 + 6 T - 9 T^{2} - 246 T^{3} - 1372 T^{4} - 246 p T^{5} - 9 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
47C2C_2 (1+6T+pT2)4 ( 1 + 6 T + p T^{2} )^{4}
53C22C_2^2 (1+98T2+p2T4)2 ( 1 + 98 T^{2} + p^{2} T^{4} )^{2}
59D4×C2D_4\times C_2 1−6T−73T2+54T3+6276T4+54pT5−73p2T6−6p3T7+p4T8 1 - 6 T - 73 T^{2} + 54 T^{3} + 6276 T^{4} + 54 p T^{5} - 73 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
61C22C_2^2 (1+7T−12T2+7pT3+p2T4)2 ( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}
67D4×C2D_4\times C_2 1−18T+127T2−1134T3+12612T4−1134pT5+127p2T6−18p3T7+p4T8 1 - 18 T + 127 T^{2} - 1134 T^{3} + 12612 T^{4} - 1134 p T^{5} + 127 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
71D4×C2D_4\times C_2 1+6T−97T2−54T3+10092T4−54pT5−97p2T6+6p3T7+p4T8 1 + 6 T - 97 T^{2} - 54 T^{3} + 10092 T^{4} - 54 p T^{5} - 97 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
73D4D_{4} (1−8T+90T2−8pT3+p2T4)2 ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1−6T+pT2)4 ( 1 - 6 T + p T^{2} )^{4}
83C2C_2 (1+4T+pT2)4 ( 1 + 4 T + p T^{2} )^{4}
89D4×C2D_4\times C_2 1−18T+97T2−882T3+14772T4−882pT5+97p2T6−18p3T7+p4T8 1 - 18 T + 97 T^{2} - 882 T^{3} + 14772 T^{4} - 882 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
97C22C_2^2 (1−9T−16T2−9pT3+p2T4)2 ( 1 - 9 T - 16 T^{2} - 9 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

−8.020247026217916835361788042620, −7.890454310843357315933530154568, −7.76604895823585456810387293220, −7.32545419090055812618387222217, −7.17435001523400540384268839752, −7.16301982154919587188416679669, −6.44681822157245728736762298299, −6.43386664599844950863617194973, −6.10644366400052375453825882071, −5.60551626322680876268635154308, −5.48198528747405048092641515389, −5.35031050011928999273122749769, −5.03758832816359361039200034102, −4.85611813310988074459383364679, −4.66746644552499408139395255332, −4.25630128671750165758581541419, −4.02518619012058607441005479869, −3.46648177626219165005588939596, −3.39697408338966575749145793737, −2.97781434624446805719630255174, −2.04460082587664742649847813553, −1.92858244175304535860187854915, −1.92262402817580827586058454024, −1.18902363818117412411488506714, −0.59675671118218918460236937320, 0.59675671118218918460236937320, 1.18902363818117412411488506714, 1.92262402817580827586058454024, 1.92858244175304535860187854915, 2.04460082587664742649847813553, 2.97781434624446805719630255174, 3.39697408338966575749145793737, 3.46648177626219165005588939596, 4.02518619012058607441005479869, 4.25630128671750165758581541419, 4.66746644552499408139395255332, 4.85611813310988074459383364679, 5.03758832816359361039200034102, 5.35031050011928999273122749769, 5.48198528747405048092641515389, 5.60551626322680876268635154308, 6.10644366400052375453825882071, 6.43386664599844950863617194973, 6.44681822157245728736762298299, 7.16301982154919587188416679669, 7.17435001523400540384268839752, 7.32545419090055812618387222217, 7.76604895823585456810387293220, 7.890454310843357315933530154568, 8.020247026217916835361788042620

Graph of the ZZ-function along the critical line