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Properties

Label 8-867e4-1.1-c0e4-0-0
Degree 88
Conductor 565036352721565036352721
Sign 11
Analytic cond. 0.03505130.0350513
Root an. cond. 0.6577910.657791
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  − 4·4-s + 4·13-s + 10·16-s − 16·52-s − 20·64-s − 4·67-s − 81-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 40·208-s + 211-s + ⋯
L(s)  = 1  − 4·4-s + 4·13-s + 10·16-s − 16·52-s − 20·64-s − 4·67-s − 81-s − 4·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 40·208-s + 211-s + ⋯

Functional equation

Λ(s)=((34⋅178)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((34⋅178)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅1783^{4} \cdot 17^{8}
Sign: 11
Analytic conductor: 0.03505130.0350513
Root analytic conductor: 0.6577910.657791
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅178, ( :0,0,0,0), 1)(8,\ 3^{4} \cdot 17^{8} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.34518903480.3451890348
L(12)L(\frac12) ≈\approx 0.34518903480.3451890348
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)
bad3C22C_2^2 1+T4 1 + T^{4}
17 1 1
good2C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
5C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
7C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
11C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
13C2C_2 (1−T+T2)4 ( 1 - T + T^{2} )^{4}
19C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
23C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
29C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
31C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
37C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
41C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
43C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
47C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
53C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
59C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
61C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
67C2C_2 (1+T+T2)4 ( 1 + T + T^{2} )^{4}
71C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
73C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
79C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
83C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
89C1C_1×\timesC1C_1 (1−T)4(1+T)4 ( 1 - T )^{4}( 1 + T )^{4}
97C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
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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

−7.82956995653123268778773059071, −7.44260067109104452478358349948, −6.93877467487005068763566155171, −6.90733675735922229648144695972, −6.55463420897253909556409351222, −5.93226144079959852517820556361, −5.92577237915978534975049110222, −5.91396263915775707123898883969, −5.81377201354538384911422407052, −5.46220063401616106482853365663, −4.96110723840541831610336365924, −4.92373286025177504971815052936, −4.74143429003153392351442594367, −4.24280479494134072127724901382, −4.12965944799308626802165964252, −3.98667549045219346519891785970, −3.87004982682175354458229525103, −3.50458912913076591424491464664, −3.20605174806432004589495505949, −3.06410198624547711167492618776, −2.73495878606900186375558057308, −1.64694741879472401837446572265, −1.36230105764151789874643651832, −1.35190921737122512040792026496, −0.69164488747293383763982657862, 0.69164488747293383763982657862, 1.35190921737122512040792026496, 1.36230105764151789874643651832, 1.64694741879472401837446572265, 2.73495878606900186375558057308, 3.06410198624547711167492618776, 3.20605174806432004589495505949, 3.50458912913076591424491464664, 3.87004982682175354458229525103, 3.98667549045219346519891785970, 4.12965944799308626802165964252, 4.24280479494134072127724901382, 4.74143429003153392351442594367, 4.92373286025177504971815052936, 4.96110723840541831610336365924, 5.46220063401616106482853365663, 5.81377201354538384911422407052, 5.91396263915775707123898883969, 5.92577237915978534975049110222, 5.93226144079959852517820556361, 6.55463420897253909556409351222, 6.90733675735922229648144695972, 6.93877467487005068763566155171, 7.44260067109104452478358349948, 7.82956995653123268778773059071

Graph of the ZZ-function along the critical line