Properties

Label 6-23e3-23.22-c2e3-0-0
Degree 66
Conductor 1216712167
Sign 11
Analytic cond. 0.2461430.246143
Root an. cond. 0.7916460.791646
Motivic weight 22
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·8-s − 69·23-s + 75·25-s − 38·27-s + 147·49-s + 78·59-s − 15·64-s − 498·101-s + 363·121-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 + 483·184-s + 191-s + 193-s + 197-s + 199-s − 525·200-s + ⋯
L(s)  = 1  − 7/8·8-s − 3·23-s + 3·25-s − 1.40·27-s + 3·49-s + 1.32·59-s − 0.234·64-s − 4.93·101-s + 3·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 21/8·184-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s − 2.62·200-s + ⋯

Functional equation

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

Invariants

Degree: 66
Conductor: 1216712167    =    23323^{3}
Sign: 11
Analytic conductor: 0.2461430.246143
Root analytic conductor: 0.7916460.791646
Motivic weight: 22
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, ( :1,1,1), 1)(6,\ 12167,\ (\ :1, 1, 1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) \approx 0.70380929510.7038092951
L(12)L(\frac12) \approx 0.70380929510.7038092951
L(2)L(2) 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 (1+pT)3 ( 1 + p T )^{3}
good2D6D_{6} 1+7T3+p6T6 1 + 7 T^{3} + p^{6} T^{6}
3D6D_{6} 1+38T3+p6T6 1 + 38 T^{3} + p^{6} T^{6}
5C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
7C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
11C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
13D6D_{6} 11082T3+p6T6 1 - 1082 T^{3} + p^{6} T^{6}
17C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
19C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
29D6D_{6} 130746T3+p6T6 1 - 30746 T^{3} + p^{6} T^{6}
31D6D_{6} 158754T3+p6T6 1 - 58754 T^{3} + p^{6} T^{6}
37C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
41D6D_{6} 143634T3+p6T6 1 - 43634 T^{3} + p^{6} T^{6}
43C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
47D6D_{6} 1+205342T3+p6T6 1 + 205342 T^{3} + p^{6} T^{6}
53C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
59C2C_2 (126T+p2T2)3 ( 1 - 26 T + p^{2} T^{2} )^{3}
61C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
67C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
71D6D_{6} 1667154T3+p6T6 1 - 667154 T^{3} + p^{6} T^{6}
73D6D_{6} 1725042T3+p6T6 1 - 725042 T^{3} + p^{6} T^{6}
79C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
83C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
89C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p T )^{3}
97C1C_1×\timesC1C_1 (1pT)3(1+pT)3 ( 1 - p T )^{3}( 1 + p 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

−16.25266090259699302544844085089, −15.38379949405769162888300515781, −15.25045957856847525792594451566, −14.82823558841975961978977417582, −14.30448934002337246511345373331, −13.87709638210915675759205960405, −13.54348074276356119184964203974, −12.88326648893801043752994094227, −12.34454102311100891116481500791, −12.18652506141296656813498136593, −11.66311862169944965457286273302, −11.12595221769137640392299636957, −10.35740039286693641585425892371, −10.33664492004809423462870333956, −9.319600292021350002126565316276, −9.255683549229959107838401341587, −8.254627656874356514129115296000, −8.249320660797426125572295542019, −7.15878206375237697440477144189, −6.81039803586467718459107985552, −5.79533420479259215020430426704, −5.67469663404156687062790353124, −4.46007887250445186823029671800, −3.69917499700457382202237665871, −2.50316094689835549482408964468, 2.50316094689835549482408964468, 3.69917499700457382202237665871, 4.46007887250445186823029671800, 5.67469663404156687062790353124, 5.79533420479259215020430426704, 6.81039803586467718459107985552, 7.15878206375237697440477144189, 8.249320660797426125572295542019, 8.254627656874356514129115296000, 9.255683549229959107838401341587, 9.319600292021350002126565316276, 10.33664492004809423462870333956, 10.35740039286693641585425892371, 11.12595221769137640392299636957, 11.66311862169944965457286273302, 12.18652506141296656813498136593, 12.34454102311100891116481500791, 12.88326648893801043752994094227, 13.54348074276356119184964203974, 13.87709638210915675759205960405, 14.30448934002337246511345373331, 14.82823558841975961978977417582, 15.25045957856847525792594451566, 15.38379949405769162888300515781, 16.25266090259699302544844085089

Graph of the ZZ-function along the critical line