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Properties

Label 12-2175e6-1.1-c0e6-0-0
Degree 1212
Conductor 1.059×10201.059\times 10^{20}
Sign 11
Analytic cond. 1.635671.63567
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s − 6·29-s − 6·41-s + 64-s + 6·81-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 + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 3·9-s − 6·29-s − 6·41-s + 64-s + 6·81-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 + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯

Functional equation

Λ(s)=((36⋅512⋅296)s/2 ΓC(s)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((36⋅512⋅296)s/2 ΓC(s)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 36⋅512⋅2963^{6} \cdot 5^{12} \cdot 29^{6}
Sign: 11
Analytic conductor: 1.635671.63567
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 36⋅512⋅296, ( :[0]6), 1)(12,\ 3^{6} \cdot 5^{12} \cdot 29^{6} ,\ ( \ : [0]^{6} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.27250158720.2725015872
L(12)L(\frac12) ≈\approx 0.27250158720.2725015872
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}
ppFp(T)F_p(T)
bad3 (1+T2)3 ( 1 + T^{2} )^{3}
5 1 1
29 (1+T)6 ( 1 + T )^{6}
good2 1−T6+T12 1 - T^{6} + T^{12}
7 1−T6+T12 1 - T^{6} + T^{12}
11 (1+T3+T6)2 ( 1 + T^{3} + T^{6} )^{2}
13 1−T6+T12 1 - T^{6} + T^{12}
17 1−T6+T12 1 - T^{6} + T^{12}
19 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
23 (1+T2)6 ( 1 + T^{2} )^{6}
31 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
37 (1+T2)6 ( 1 + T^{2} )^{6}
41 (1+T+T2)6 ( 1 + T + T^{2} )^{6}
43 (1+T2)6 ( 1 + T^{2} )^{6}
47 1−T6+T12 1 - T^{6} + T^{12}
53 (1+T2)6 ( 1 + T^{2} )^{6}
59 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
61 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
67 1−T6+T12 1 - T^{6} + T^{12}
71 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
73 (1+T2)6 ( 1 + T^{2} )^{6}
79 (1−T)6(1+T)6 ( 1 - T )^{6}( 1 + T )^{6}
83 (1+T2)6 ( 1 + T^{2} )^{6}
89 (1−T3+T6)2 ( 1 - T^{3} + T^{6} )^{2}
97 (1+T2)6 ( 1 + T^{2} )^{6}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.02567961140049885902652765560, −4.93310556136923982843168318248, −4.87521113036297526372731130831, −4.54052864524554951160963920347, −4.18238621047646188466607106595, −3.99792092604343512027324767418, −3.92227413147199346103681984728, −3.92038011284746429395361893673, −3.68358561489941601004838613502, −3.43410170924670516091633517555, −3.41521212120751373256652689672, −3.37782493602904219011034578546, −3.08517586013256609681273757950, −2.89900020872895979362590543461, −2.74694215368942342595023696109, −2.63092421854304962046399015711, −2.38606741449672637671138149564, −2.06142696728767120350601762134, −1.88957024790543485481415868619, −1.79161249331612592450483384027, −1.78495838208498490317553901161, −1.56395519216789662685669275993, −1.17871366093912010891378725131, −0.52974695487465657582815957331, −0.30197695048536623300829006736, 0.30197695048536623300829006736, 0.52974695487465657582815957331, 1.17871366093912010891378725131, 1.56395519216789662685669275993, 1.78495838208498490317553901161, 1.79161249331612592450483384027, 1.88957024790543485481415868619, 2.06142696728767120350601762134, 2.38606741449672637671138149564, 2.63092421854304962046399015711, 2.74694215368942342595023696109, 2.89900020872895979362590543461, 3.08517586013256609681273757950, 3.37782493602904219011034578546, 3.41521212120751373256652689672, 3.43410170924670516091633517555, 3.68358561489941601004838613502, 3.92038011284746429395361893673, 3.92227413147199346103681984728, 3.99792092604343512027324767418, 4.18238621047646188466607106595, 4.54052864524554951160963920347, 4.87521113036297526372731130831, 4.93310556136923982843168318248, 5.02567961140049885902652765560

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.