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Properties

Label 10-1639e5-1639.1638-c0e5-0-0
Degree 1010
Conductor 1.183×10161.183\times 10^{16}
Sign 11
Analytic cond. 0.3661680.366168
Root an. cond. 0.9044150.904415
Motivic weight 00
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-s − 5-s + 5·9-s − 10-s − 5·11-s + 13-s + 5·18-s − 5·22-s + 26-s − 31-s − 37-s + 41-s + 43-s − 5·45-s − 47-s + 5·49-s − 53-s + 5·55-s − 62-s − 65-s − 67-s − 74-s + 79-s + 15·81-s + 82-s + 83-s + 86-s + ⋯
L(s)  = 1  + 2-s − 5-s + 5·9-s − 10-s − 5·11-s + 13-s + 5·18-s − 5·22-s + 26-s − 31-s − 37-s + 41-s + 43-s − 5·45-s − 47-s + 5·49-s − 53-s + 5·55-s − 62-s − 65-s − 67-s − 74-s + 79-s + 15·81-s + 82-s + 83-s + 86-s + ⋯

Functional equation

Λ(s)=((115⋅1495)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 149^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((115⋅1495)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{5} \cdot 149^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1010
Conductor: 115⋅149511^{5} \cdot 149^{5}
Sign: 11
Analytic conductor: 0.3661680.366168
Root analytic conductor: 0.9044150.904415
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ1639(1638,⋅)\chi_{1639} (1638, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (10, 115⋅1495, ( :0,0,0,0,0), 1)(10,\ 11^{5} \cdot 149^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8121861561.812186156
L(12)L(\frac12) ≈\approx 1.8121861561.812186156
L(1)L(1) 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)
bad11C1C_1 (1+T)5 ( 1 + T )^{5}
149C1C_1 (1+T)5 ( 1 + T )^{5}
good2C10C_{10} 1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}
3C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
5C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
7C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
13C10C_{10} 1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}
17C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
19C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
23C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
29C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
31C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
37C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
41C10C_{10} 1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}
43C10C_{10} 1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}
47C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
53C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
59C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
61C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
67C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
71C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
73C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
79C10C_{10} 1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}
83C10C_{10} 1−T+T2−T3+T4−T5+T6−T7+T8−T9+T10 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10}
89C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
97C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
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   L(s)=∏p ∏j=110(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.86333919393953720712630632781, −5.62797145858596236717231223954, −5.20587205014181923414961669936, −5.16523614008019721986931481547, −5.11513765964475771141380518312, −5.00459954430413315676488460533, −4.54684242912588496141258037579, −4.52226048389039171857772780611, −4.50926597363829723951203857267, −4.50855172360119910455989588297, −3.85533486886069327681159816252, −3.82586986607491225422158413863, −3.78977117293888369663500645147, −3.73007687175564729836919657127, −3.41687932666789020949594025352, −3.10721616634446961088516680114, −2.62503090041648857293157840730, −2.60574990946182001377128292025, −2.50111511526238545857425774303, −2.08574379024668778768221950481, −1.87058245062242554533984707957, −1.82687078923438995194039661081, −1.26586894372916836469215692354, −1.01891704136315221620127410073, −0.67966963927129155411086248780, 0.67966963927129155411086248780, 1.01891704136315221620127410073, 1.26586894372916836469215692354, 1.82687078923438995194039661081, 1.87058245062242554533984707957, 2.08574379024668778768221950481, 2.50111511526238545857425774303, 2.60574990946182001377128292025, 2.62503090041648857293157840730, 3.10721616634446961088516680114, 3.41687932666789020949594025352, 3.73007687175564729836919657127, 3.78977117293888369663500645147, 3.82586986607491225422158413863, 3.85533486886069327681159816252, 4.50855172360119910455989588297, 4.50926597363829723951203857267, 4.52226048389039171857772780611, 4.54684242912588496141258037579, 5.00459954430413315676488460533, 5.11513765964475771141380518312, 5.16523614008019721986931481547, 5.20587205014181923414961669936, 5.62797145858596236717231223954, 5.86333919393953720712630632781

Graph of the ZZ-function along the critical line