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Properties

Label 12-2640e6-1.1-c1e6-0-1
Degree 1212
Conductor 3.386×10203.386\times 10^{20}
Sign 11
Analytic cond. 8.77578×1078.77578\times 10^{7}
Root an. cond. 4.591354.59135
Motivic weight 11
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  + 2·5-s − 3·9-s + 6·11-s − 4·19-s + 25-s + 16·29-s − 16·31-s − 16·41-s − 6·45-s + 10·49-s + 12·55-s − 16·59-s + 4·61-s + 24·71-s − 12·79-s + 6·81-s − 4·89-s − 8·95-s − 18·99-s − 80·101-s + 12·109-s + 21·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s + 1.80·11-s − 0.917·19-s + 1/5·25-s + 2.97·29-s − 2.87·31-s − 2.49·41-s − 0.894·45-s + 10/7·49-s + 1.61·55-s − 2.08·59-s + 0.512·61-s + 2.84·71-s − 1.35·79-s + 2/3·81-s − 0.423·89-s − 0.820·95-s − 1.80·99-s − 7.96·101-s + 1.14·109-s + 1.90·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=((224⋅36⋅56⋅116)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅36⋅56⋅116)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 224⋅36⋅56⋅1162^{24} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6}
Sign: 11
Analytic conductor: 8.77578×1078.77578\times 10^{7}
Root analytic conductor: 4.591354.59135
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 224⋅36⋅56⋅116, ( :[1/2]6), 1)(12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 11^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.5645741263.564574126
L(12)L(\frac12) ≈\approx 3.5645741263.564574126
L(32)L(\frac{3}{2}) 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 1
3 (1+T2)3 ( 1 + T^{2} )^{3}
5 1−2T+3T2−12T3+3pT4−2p2T5+p3T6 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
11 (1−T)6 ( 1 - T )^{6}
good7 1−10T2+31T4−124T6+31p2T8−10p4T10+p6T12 1 - 10 T^{2} + 31 T^{4} - 124 T^{6} + 31 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12}
13 1+14T2+311T4+4692T6+311p2T8+14p4T10+p6T12 1 + 14 T^{2} + 311 T^{4} + 4692 T^{6} + 311 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12}
17 1−30T2+1151T4−18164T6+1151p2T8−30p4T10+p6T12 1 - 30 T^{2} + 1151 T^{4} - 18164 T^{6} + 1151 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12}
19 (1+2T+5T2−108T3+5pT4+2p2T5+p3T6)2 ( 1 + 2 T + 5 T^{2} - 108 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
23 (1−30T2+p2T4)3 ( 1 - 30 T^{2} + p^{2} T^{4} )^{3}
29 (1−8T+3pT2−432T3+3p2T4−8p2T5+p3T6)2 ( 1 - 8 T + 3 p T^{2} - 432 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
31 (1+8T+101T2+480T3+101pT4+8p2T5+p3T6)2 ( 1 + 8 T + 101 T^{2} + 480 T^{3} + 101 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 1−174T2+13943T4−655652T6+13943p2T8−174p4T10+p6T12 1 - 174 T^{2} + 13943 T^{4} - 655652 T^{6} + 13943 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12}
41 (1+8T+3pT2+624T3+3p2T4+8p2T5+p3T6)2 ( 1 + 8 T + 3 p T^{2} + 624 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 1−226T2+22423T4−1251628T6+22423p2T8−226p4T10+p6T12 1 - 226 T^{2} + 22423 T^{4} - 1251628 T^{6} + 22423 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12}
47 1−186T2+17903T4−1043180T6+17903p2T8−186p4T10+p6T12 1 - 186 T^{2} + 17903 T^{4} - 1043180 T^{6} + 17903 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12}
53 1−190T2+20375T4−1316100T6+20375p2T8−190p4T10+p6T12 1 - 190 T^{2} + 20375 T^{4} - 1316100 T^{6} + 20375 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12}
59 (1+8T+113T2+1024T3+113pT4+8p2T5+p3T6)2 ( 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1−2T+131T2−204T3+131pT4−2p2T5+p3T6)2 ( 1 - 2 T + 131 T^{2} - 204 T^{3} + 131 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 1−226T2+28103T4−2235900T6+28103p2T8−226p4T10+p6T12 1 - 226 T^{2} + 28103 T^{4} - 2235900 T^{6} + 28103 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12}
71 (1−12T+245T2−1688T3+245pT4−12p2T5+p3T6)2 ( 1 - 12 T + 245 T^{2} - 1688 T^{3} + 245 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−234T2+27279T4−2258332T6+27279p2T8−234p4T10+p6T12 1 - 234 T^{2} + 27279 T^{4} - 2258332 T^{6} + 27279 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12}
79 (1+6T+233T2+940T3+233pT4+6p2T5+p3T6)2 ( 1 + 6 T + 233 T^{2} + 940 T^{3} + 233 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 1−486T2+99383T4−10945028T6+99383p2T8−486p4T10+p6T12 1 - 486 T^{2} + 99383 T^{4} - 10945028 T^{6} + 99383 p^{2} T^{8} - 486 p^{4} T^{10} + p^{6} T^{12}
89 (1+2T+255T2+348T3+255pT4+2p2T5+p3T6)2 ( 1 + 2 T + 255 T^{2} + 348 T^{3} + 255 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
97 1−454T2+94543T4−11606932T6+94543p2T8−454p4T10+p6T12 1 - 454 T^{2} + 94543 T^{4} - 11606932 T^{6} + 94543 p^{2} T^{8} - 454 p^{4} T^{10} + p^{6} T^{12}
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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

−4.46594944128980453040777266346, −4.42885752643919455481606105279, −4.39892290537778238486795716724, −4.19774384854639048745462260440, −3.90567108098387751749639048552, −3.86091782906617814168235339385, −3.65507327557211832922316719412, −3.52303173437102395124376152919, −3.50543622828446913261807195992, −3.25956660427373176245916677486, −2.97215898817102099861262058599, −2.90823864962864757402690948332, −2.77514270505209182235749349670, −2.42771267176284252716214710961, −2.40387526881758131931601682775, −2.39371914054203298518144493601, −1.91266082906115039153081005636, −1.77532473541198652605608151932, −1.72359574976105642656954148587, −1.46287185118712345549711804443, −1.28855809257500958653215153678, −1.10955926487437224064424706370, −0.841842234192493177452283137314, −0.34006363423594707186063484846, −0.27579647668228536578355633447, 0.27579647668228536578355633447, 0.34006363423594707186063484846, 0.841842234192493177452283137314, 1.10955926487437224064424706370, 1.28855809257500958653215153678, 1.46287185118712345549711804443, 1.72359574976105642656954148587, 1.77532473541198652605608151932, 1.91266082906115039153081005636, 2.39371914054203298518144493601, 2.40387526881758131931601682775, 2.42771267176284252716214710961, 2.77514270505209182235749349670, 2.90823864962864757402690948332, 2.97215898817102099861262058599, 3.25956660427373176245916677486, 3.50543622828446913261807195992, 3.52303173437102395124376152919, 3.65507327557211832922316719412, 3.86091782906617814168235339385, 3.90567108098387751749639048552, 4.19774384854639048745462260440, 4.39892290537778238486795716724, 4.42885752643919455481606105279, 4.46594944128980453040777266346

Graph of the ZZ-function along the critical line

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