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Properties

Label 6-71e3-71.70-c0e3-0-0
Degree 66
Conductor 357911357911
Sign 11
Analytic cond. 4.44883×10−54.44883\times 10^{-5}
Root an. cond. 0.1882380.188238
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 − 3-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯
L(s)  = 1  − 2-s − 3-s − 5-s + 6-s + 10-s + 15-s − 19-s − 29-s − 30-s − 37-s + 38-s − 43-s + 3·49-s + 57-s + 58-s + 3·71-s − 73-s + 74-s − 79-s − 83-s + 86-s + 87-s − 89-s + 95-s − 3·98-s − 101-s − 103-s + ⋯

Functional equation

Λ(s)=(357911s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(357911s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 357911 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 66
Conductor: 357911357911    =    71371^{3}
Sign: 11
Analytic conductor: 4.44883×10−54.44883\times 10^{-5}
Root analytic conductor: 0.1882380.188238
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ71(70,⋅)\chi_{71} (70, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 357911, ( :0,0,0), 1)(6,\ 357911,\ (\ :0, 0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.054982234800.05498223480
L(12)L(\frac12) ≈\approx 0.054982234800.05498223480
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)
bad71C1C_1 (1−T)3 ( 1 - T )^{3}
good2C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
3C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
5C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
7C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
11C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
13C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
17C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
19C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
23C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
29C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
31C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
37C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
41C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
43C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
47C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
53C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
59C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
61C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
67C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
73C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
79C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
83C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
89C6C_6 1+T+T2+T3+T4+T5+T6 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}
97C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
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   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.58399519906890803499351778176, −12.88778415189161766638407934767, −12.62842911673974380337261507666, −12.42351194541329992426458232503, −11.75493220562099759167484344236, −11.67985104016781996899636064300, −11.17081547941659221637655464575, −11.05204135964904542931077288526, −10.37263061263202496602275828991, −10.24785144415084892602116520791, −9.646341880691708918339787590058, −9.249596659145116242975209696115, −8.775621347957351956553642935712, −8.392664296558968410521780245040, −8.308375254782647564178629435651, −7.34866153009380292102541584258, −7.32935458632589252277476848657, −6.76877928720481535930982673330, −6.02203245587172561884189386059, −5.76082408647750230290643880271, −5.19361966854786324344583978535, −4.51844117616935213608968885293, −3.95531641366854447656333489742, −3.39979849702102685652285544820, −2.19448912992332732423794638884, 2.19448912992332732423794638884, 3.39979849702102685652285544820, 3.95531641366854447656333489742, 4.51844117616935213608968885293, 5.19361966854786324344583978535, 5.76082408647750230290643880271, 6.02203245587172561884189386059, 6.76877928720481535930982673330, 7.32935458632589252277476848657, 7.34866153009380292102541584258, 8.308375254782647564178629435651, 8.392664296558968410521780245040, 8.775621347957351956553642935712, 9.249596659145116242975209696115, 9.646341880691708918339787590058, 10.24785144415084892602116520791, 10.37263061263202496602275828991, 11.05204135964904542931077288526, 11.17081547941659221637655464575, 11.67985104016781996899636064300, 11.75493220562099759167484344236, 12.42351194541329992426458232503, 12.62842911673974380337261507666, 12.88778415189161766638407934767, 13.58399519906890803499351778176

Graph of the ZZ-function along the critical line