Properties

Label 6-23e3-23.22-c8e3-0-0
Degree 66
Conductor 1216712167
Sign 11
Analytic cond. 822.580822.580
Root an. cond. 3.060993.06099
Motivic weight 88
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  − 1.95e3·8-s + 8.39e5·23-s + 1.17e6·25-s − 1.06e6·27-s + 1.72e7·49-s + 4.58e7·59-s − 1.29e7·64-s − 4.70e8·101-s + 6.43e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s − 1.63e9·184-s + 191-s + 193-s + 197-s + 199-s − 2.28e9·200-s + ⋯
L(s)  = 1  − 0.476·8-s + 3·23-s + 3·25-s − 1.99·27-s + 3·49-s + 3.78·59-s − 0.773·64-s − 4.52·101-s + 3·121-s − 1.42·184-s − 1.42·200-s + 0.952·216-s + ⋯

Functional equation

Λ(s)=(12167s/2ΓC(s)3L(s)=(Λ(9s)\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
Λ(s)=(12167s/2ΓC(s+4)3L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 12167 ^{s/2} \, \Gamma_{\C}(s+4)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 66
Conductor: 1216712167    =    23323^{3}
Sign: 11
Analytic conductor: 822.580822.580
Root analytic conductor: 3.060993.06099
Motivic weight: 88
Rational: yes
Arithmetic: yes
Character: induced by χ23(22,)\chi_{23} (22, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 12167, ( :4,4,4), 1)(6,\ 12167,\ (\ :4, 4, 4),\ 1)

Particular Values

L(92)L(\frac{9}{2}) \approx 3.1010808803.101080880
L(12)L(\frac12) \approx 3.1010808803.101080880
L(5)L(5) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad23C1C_1 (1p4T)3 ( 1 - p^{4} T )^{3}
good2D6D_{6} 1+1951T3+p24T6 1 + 1951 T^{3} + p^{24} T^{6}
3D6D_{6} 1+1062686T3+p24T6 1 + 1062686 T^{3} + p^{24} T^{6}
5C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
7C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
11C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
13D6D_{6} 125363320370274T3+p24T6 1 - 25363320370274 T^{3} + p^{24} T^{6}
17C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
19C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
29D6D_{6} 1+647932355939762206T3+p24T6 1 + 647932355939762206 T^{3} + p^{24} T^{6}
31D6D_{6} 11237087799571624194T3+p24T6 1 - 1237087799571624194 T^{3} + p^{24} T^{6}
37C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
41D6D_{6} 112576527614080568514T3+p24T6 1 - 12576527614080568514 T^{3} + p^{24} T^{6}
43C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
47D6D_{6} 1 1 - 19 ⁣ ⁣5419\!\cdots\!54T3+p24T6 T^{3} + p^{24} T^{6}
53C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
59C2C_2 (115279074T+p8T2)3 ( 1 - 15279074 T + p^{8} T^{2} )^{3}
61C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
67C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
71D6D_{6} 1 1 - 28 ⁣ ⁣9428\!\cdots\!94T3+p24T6 T^{3} + p^{24} T^{6}
73D6D_{6} 1 1 - 39 ⁣ ⁣5439\!\cdots\!54T3+p24T6 T^{3} + p^{24} T^{6}
79C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
83C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
89C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
97C1C_1×\timesC1C_1 (1p4T)3(1+p4T)3 ( 1 - p^{4} T )^{3}( 1 + p^{4} T )^{3}
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   L(s)=p j=16(1αj,pps)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

−14.60505482600838178979882243183, −13.57754776366493196214640179035, −13.57358699137694629103351250428, −12.94340974939646264484421352028, −12.58994809827745075575670473695, −12.20573812798288630229092135669, −11.48934113174410361454083235145, −11.13020128337881722518285137089, −10.78407804078334704513648951630, −10.26071600882990030330086952673, −9.569195258661682746768575480396, −9.047364399505374511119356771709, −8.785528370880907108738112345945, −8.312193720324238660650046295629, −7.22331694672823028526268534820, −7.12821196931798717727353152062, −6.61290661913085672954290949270, −5.48365710965517680196720795192, −5.41807275592759405744549257376, −4.54563824817196368509283972522, −3.74241792710594257007497295027, −2.95312795013608813421629405734, −2.44985366376325530211392602373, −1.15438384073458975501791693746, −0.66124939014099614407429649788, 0.66124939014099614407429649788, 1.15438384073458975501791693746, 2.44985366376325530211392602373, 2.95312795013608813421629405734, 3.74241792710594257007497295027, 4.54563824817196368509283972522, 5.41807275592759405744549257376, 5.48365710965517680196720795192, 6.61290661913085672954290949270, 7.12821196931798717727353152062, 7.22331694672823028526268534820, 8.312193720324238660650046295629, 8.785528370880907108738112345945, 9.047364399505374511119356771709, 9.569195258661682746768575480396, 10.26071600882990030330086952673, 10.78407804078334704513648951630, 11.13020128337881722518285137089, 11.48934113174410361454083235145, 12.20573812798288630229092135669, 12.58994809827745075575670473695, 12.94340974939646264484421352028, 13.57358699137694629103351250428, 13.57754776366493196214640179035, 14.60505482600838178979882243183

Graph of the ZZ-function along the critical line