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Properties

Label 6-2175e3-87.86-c0e3-0-3
Degree 66
Conductor 1028910937510289109375
Sign 11
Analytic cond. 1.278931.27893
Root an. cond. 1.041851.04185
Motivic weight 00
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  + 3·3-s − 8-s + 6·9-s − 3·24-s + 10·27-s + 3·29-s − 3·41-s − 6·72-s + 15·81-s + 9·87-s − 3·103-s − 9·123-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 + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 3·3-s − 8-s + 6·9-s − 3·24-s + 10·27-s + 3·29-s − 3·41-s − 6·72-s + 15·81-s + 9·87-s − 3·103-s − 9·123-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 + 191-s + 193-s + 197-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅293)s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((33⋅56⋅293)s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅2933^{3} \cdot 5^{6} \cdot 29^{3}
Sign: 11
Analytic conductor: 1.278931.27893
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ2175(1826,⋅)\chi_{2175} (1826, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 33⋅56⋅293, ( :0,0,0), 1)(6,\ 3^{3} \cdot 5^{6} \cdot 29^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 4.5151330464.515133046
L(12)L(\frac12) ≈\approx 4.5151330464.515133046
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)
bad3C1C_1 (1−T)3 ( 1 - T )^{3}
5 1 1
29C1C_1 (1−T)3 ( 1 - T )^{3}
good2C6C_6 1+T3+T6 1 + T^{3} + T^{6}
7C6C_6 1+T3+T6 1 + T^{3} + T^{6}
11C6C_6 1+T3+T6 1 + T^{3} + T^{6}
13C6C_6 1+T3+T6 1 + T^{3} + T^{6}
17C6C_6 1+T3+T6 1 + T^{3} + T^{6}
19C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
23C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
31C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
37C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
41C2C_2 (1+T+T2)3 ( 1 + T + T^{2} )^{3}
43C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
47C6C_6 1+T3+T6 1 + T^{3} + T^{6}
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}
67C6C_6 1+T3+T6 1 + T^{3} + T^{6}
71C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
73C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
79C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
83C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
89C6C_6 1+T3+T6 1 + T^{3} + 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

−8.408644787817091106536694094947, −7.991267099831890594699457123940, −7.86389041499769093945870291767, −7.66275555624468185180681167655, −7.31490326498810007029635375582, −6.80849427207487440746421475832, −6.72796410866918711335748441382, −6.68791974876211866439026023359, −6.34545213277789513529148227185, −6.02107150574453049313613739909, −5.33594900268624110175744969100, −5.19004193341438494320000767310, −4.93788023362431131508179829057, −4.47194071957267156608671701903, −4.33743472566190014343759443314, −3.98505075658530088173752754451, −3.54680228661875330699676854351, −3.49223951512014967179331160779, −2.94212328958274825648708830591, −2.83614782990033014927984572165, −2.74687679771882961240533101605, −2.13497346391147626444047201176, −1.93764703926138004375136775211, −1.25939835582413232369720352277, −1.18432516009006482961781076353, 1.18432516009006482961781076353, 1.25939835582413232369720352277, 1.93764703926138004375136775211, 2.13497346391147626444047201176, 2.74687679771882961240533101605, 2.83614782990033014927984572165, 2.94212328958274825648708830591, 3.49223951512014967179331160779, 3.54680228661875330699676854351, 3.98505075658530088173752754451, 4.33743472566190014343759443314, 4.47194071957267156608671701903, 4.93788023362431131508179829057, 5.19004193341438494320000767310, 5.33594900268624110175744969100, 6.02107150574453049313613739909, 6.34545213277789513529148227185, 6.68791974876211866439026023359, 6.72796410866918711335748441382, 6.80849427207487440746421475832, 7.31490326498810007029635375582, 7.66275555624468185180681167655, 7.86389041499769093945870291767, 7.991267099831890594699457123940, 8.408644787817091106536694094947

Graph of the ZZ-function along the critical line