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Properties

Label 32-772e16-1.1-c0e16-0-0
Degree 3232
Conductor 1.592×10461.592\times 10^{46}
Sign 11
Analytic cond. 2.35716×10−72.35716\times 10^{-7}
Root an. cond. 0.6207070.620707
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  − 8·17-s + 8·49-s − 8·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 241-s + ⋯
L(s)  = 1  − 8·17-s + 8·49-s − 8·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 241-s + ⋯

Functional equation

Λ(s)=((232⋅19316)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((232⋅19316)s/2 ΓC(s)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 232⋅193162^{32} \cdot 193^{16}
Sign: 11
Analytic conductor: 2.35716×10−72.35716\times 10^{-7}
Root analytic conductor: 0.6207070.620707
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 232⋅19316, ( :[0]16), 1)(32,\ 2^{32} \cdot 193^{16} ,\ ( \ : [0]^{16} ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.16042332550.1604233255
L(12)L(\frac12) ≈\approx 0.16042332550.1604233255
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)
bad2 1−T8+T16 1 - T^{8} + T^{16}
193 (1+T4)4 ( 1 + T^{4} )^{4}
good3 (1+T4)8 ( 1 + T^{4} )^{8}
5 (1−T4+T8)2(1−T8+T16) ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} )
7 (1−T2+T4)8 ( 1 - T^{2} + T^{4} )^{8}
11 (1+T16)2 ( 1 + T^{16} )^{2}
13 (1−T2+T4)4(1−T8+T16) ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{8} + T^{16} )
17 (1+T+T2)8(1−T8+T16) ( 1 + T + T^{2} )^{8}( 1 - T^{8} + T^{16} )
19 1−T16+T32 1 - T^{16} + T^{32}
23 (1+T8)4 ( 1 + T^{8} )^{4}
29 (1−T2+T4)4(1−T8+T16) ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{8} + T^{16} )
31 (1−T8+T16)2 ( 1 - T^{8} + T^{16} )^{2}
37 (1−T4+T8)2(1+T8)2 ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2}
41 (1+T4)4(1−T8+T16) ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} )
43 (1+T2)16 ( 1 + T^{2} )^{16}
47 1−T16+T32 1 - T^{16} + T^{32}
53 (1+T4)4(1−T8+T16) ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} )
59 (1−T4+T8)4 ( 1 - T^{4} + T^{8} )^{4}
61 (1−T4+T8)2(1+T8)2 ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2}
67 (1+T8)4 ( 1 + T^{8} )^{4}
71 (1+T16)2 ( 1 + T^{16} )^{2}
73 (1+T4)4(1−T8+T16) ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} )
79 1−T16+T32 1 - T^{16} + T^{32}
83 (1−T8+T16)2 ( 1 - T^{8} + T^{16} )^{2}
89 (1+T2)8(1+T8)2 ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2}
97 (1+T8)2(1−T8+T16) ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} )
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   L(s)=∏p ∏j=132(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−2.90060114328725907837314534759, −2.81878817228431584719077475117, −2.79033598182948095848059520565, −2.74704202351608846512647275955, −2.70454663379331532259835206004, −2.70431095472434333458226947391, −2.69835023649434470594164621772, −2.40991930193802078733944442266, −2.40978064899841668991239670395, −2.30372572472568799329027068713, −2.19888845135035083034594986170, −2.17804168817621273354707273559, −1.90924306303309231507689577665, −1.89005802164656225508777371854, −1.84974517055905239695531485583, −1.79641561234089062297370619511, −1.77458625032616306547956977416, −1.69640654096583928862495610121, −1.68730861585721120479540025380, −1.30861616326027077854526527936, −1.14505044608192245189447664147, −1.10002274406697028666163995662, −0.864641721165017708832117213964, −0.792217138681178982106963346258, −0.55502463297254491571726824398, 0.55502463297254491571726824398, 0.792217138681178982106963346258, 0.864641721165017708832117213964, 1.10002274406697028666163995662, 1.14505044608192245189447664147, 1.30861616326027077854526527936, 1.68730861585721120479540025380, 1.69640654096583928862495610121, 1.77458625032616306547956977416, 1.79641561234089062297370619511, 1.84974517055905239695531485583, 1.89005802164656225508777371854, 1.90924306303309231507689577665, 2.17804168817621273354707273559, 2.19888845135035083034594986170, 2.30372572472568799329027068713, 2.40978064899841668991239670395, 2.40991930193802078733944442266, 2.69835023649434470594164621772, 2.70431095472434333458226947391, 2.70454663379331532259835206004, 2.74704202351608846512647275955, 2.79033598182948095848059520565, 2.81878817228431584719077475117, 2.90060114328725907837314534759

Graph of the ZZ-function along the critical line

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