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Properties

Label 12-5408e6-1.1-c1e6-0-2
Degree 1212
Conductor 2.502×10222.502\times 10^{22}
Sign 11
Analytic cond. 6.48459×1096.48459\times 10^{9}
Root an. cond. 6.571386.57138
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 66

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s − 3·9-s + 6·11-s − 6·17-s − 6·19-s − 12·25-s + 6·29-s − 24·31-s + 12·47-s + 9·49-s − 24·53-s − 6·59-s − 18·61-s + 18·63-s − 30·67-s − 18·71-s − 36·77-s − 6·81-s − 36·83-s − 18·99-s − 18·101-s + 30·113-s + 36·119-s − 15·121-s + 127-s + 131-s + 36·133-s + ⋯
L(s)  = 1  − 2.26·7-s − 9-s + 1.80·11-s − 1.45·17-s − 1.37·19-s − 2.39·25-s + 1.11·29-s − 4.31·31-s + 1.75·47-s + 9/7·49-s − 3.29·53-s − 0.781·59-s − 2.30·61-s + 2.26·63-s − 3.66·67-s − 2.13·71-s − 4.10·77-s − 2/3·81-s − 3.95·83-s − 1.80·99-s − 1.79·101-s + 2.82·113-s + 3.30·119-s − 1.36·121-s + 0.0887·127-s + 0.0873·131-s + 3.12·133-s + ⋯

Functional equation

Λ(s)=((230⋅1312)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((230⋅1312)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 230⋅13122^{30} \cdot 13^{12}
Sign: 11
Analytic conductor: 6.48459×1096.48459\times 10^{9}
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 66
Selberg data: (12, 230⋅1312, ( :[1/2]6), 1)(12,\ 2^{30} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
13 1 1
good3 1+pT2+5pT4+14pT6+5p3T8+p5T10+p6T12 1 + p T^{2} + 5 p T^{4} + 14 p T^{6} + 5 p^{3} T^{8} + p^{5} T^{10} + p^{6} T^{12}
5 1+12T2+48T4+118T6+48p2T8+12p4T10+p6T12 1 + 12 T^{2} + 48 T^{4} + 118 T^{6} + 48 p^{2} T^{8} + 12 p^{4} T^{10} + p^{6} T^{12}
7 (1+3T+9T2+26T3+9pT4+3p2T5+p3T6)2 ( 1 + 3 T + 9 T^{2} + 26 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
11 (1−3T+21T2−50T3+21pT4−3p2T5+p3T6)2 ( 1 - 3 T + 21 T^{2} - 50 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
17 (1+T+pT2)6 ( 1 + T + p T^{2} )^{6}
19 (1+3T+39T2+78T3+39pT4+3p2T5+p3T6)2 ( 1 + 3 T + 39 T^{2} + 78 T^{3} + 39 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
23 1+3T2+327T4−9926T6+327p2T8+3p4T10+p6T12 1 + 3 T^{2} + 327 T^{4} - 9926 T^{6} + 327 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12}
29 (1−3T+66T2−143T3+66pT4−3p2T5+p3T6)2 ( 1 - 3 T + 66 T^{2} - 143 T^{3} + 66 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
31 (1+12T+117T2+720T3+117pT4+12p2T5+p3T6)2 ( 1 + 12 T + 117 T^{2} + 720 T^{3} + 117 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 1+141T2+9822T4+437033T6+9822p2T8+141p4T10+p6T12 1 + 141 T^{2} + 9822 T^{4} + 437033 T^{6} + 9822 p^{2} T^{8} + 141 p^{4} T^{10} + p^{6} T^{12}
41 1+165T2+13350T4+671185T6+13350p2T8+165p4T10+p6T12 1 + 165 T^{2} + 13350 T^{4} + 671185 T^{6} + 13350 p^{2} T^{8} + 165 p^{4} T^{10} + p^{6} T^{12}
43 1+159T2+13395T4+710714T6+13395p2T8+159p4T10+p6T12 1 + 159 T^{2} + 13395 T^{4} + 710714 T^{6} + 13395 p^{2} T^{8} + 159 p^{4} T^{10} + p^{6} T^{12}
47 (1−6T+105T2−356T3+105pT4−6p2T5+p3T6)2 ( 1 - 6 T + 105 T^{2} - 356 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
53 (1+12T+186T2+1268T3+186pT4+12p2T5+p3T6)2 ( 1 + 12 T + 186 T^{2} + 1268 T^{3} + 186 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
59 (1+3T+105T2+34T3+105pT4+3p2T5+p3T6)2 ( 1 + 3 T + 105 T^{2} + 34 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1+9T+162T2+861T3+162pT4+9p2T5+p3T6)2 ( 1 + 9 T + 162 T^{2} + 861 T^{3} + 162 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 (1+15T+171T2+1222T3+171pT4+15p2T5+p3T6)2 ( 1 + 15 T + 171 T^{2} + 1222 T^{3} + 171 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} )^{2}
71 (1+9T+219T2+1258T3+219pT4+9p2T5+p3T6)2 ( 1 + 9 T + 219 T^{2} + 1258 T^{3} + 219 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1+204T2+27696T4+2359118T6+27696p2T8+204p4T10+p6T12 1 + 204 T^{2} + 27696 T^{4} + 2359118 T^{6} + 27696 p^{2} T^{8} + 204 p^{4} T^{10} + p^{6} T^{12}
79 1+78T2+11487T4+1040996T6+11487p2T8+78p4T10+p6T12 1 + 78 T^{2} + 11487 T^{4} + 1040996 T^{6} + 11487 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12}
83 (1+18T+297T2+2860T3+297pT4+18p2T5+p3T6)2 ( 1 + 18 T + 297 T^{2} + 2860 T^{3} + 297 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2}
89 1+99T2+21795T4+1563778T6+21795p2T8+99p4T10+p6T12 1 + 99 T^{2} + 21795 T^{4} + 1563778 T^{6} + 21795 p^{2} T^{8} + 99 p^{4} T^{10} + p^{6} T^{12}
97 1+267T2+38355T4+4179602T6+38355p2T8+267p4T10+p6T12 1 + 267 T^{2} + 38355 T^{4} + 4179602 T^{6} + 38355 p^{2} T^{8} + 267 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

−4.48766409520434107982067950529, −4.34258407000099000504783092455, −4.32931945686186841540505783581, −4.16109199379403991669064849817, −4.13927376154044942543925369011, −3.83811994009846207999911935253, −3.80173021184388771907555792360, −3.51996184189447659278502805493, −3.39952991168740037849350365139, −3.33143201013295558085777338226, −3.21854650908424466760777774395, −3.15266693306383504762731323887, −3.11557953033200559250174127929, −2.75496763006946786643693295169, −2.55027925300090534891806779363, −2.41083640231504220468308128770, −2.31324370298039387205154172010, −2.19925141910419266438701412431, −2.10043861106555732281903691750, −1.65270321050962222121567897004, −1.63511445230337642103434733710, −1.37328935951377395597195585054, −1.36973162213030659734144985095, −1.21544427435691185965417242963, −1.01476566759720851377389782471, 0, 0, 0, 0, 0, 0, 1.01476566759720851377389782471, 1.21544427435691185965417242963, 1.36973162213030659734144985095, 1.37328935951377395597195585054, 1.63511445230337642103434733710, 1.65270321050962222121567897004, 2.10043861106555732281903691750, 2.19925141910419266438701412431, 2.31324370298039387205154172010, 2.41083640231504220468308128770, 2.55027925300090534891806779363, 2.75496763006946786643693295169, 3.11557953033200559250174127929, 3.15266693306383504762731323887, 3.21854650908424466760777774395, 3.33143201013295558085777338226, 3.39952991168740037849350365139, 3.51996184189447659278502805493, 3.80173021184388771907555792360, 3.83811994009846207999911935253, 4.13927376154044942543925369011, 4.16109199379403991669064849817, 4.32931945686186841540505783581, 4.34258407000099000504783092455, 4.48766409520434107982067950529

Graph of the ZZ-function along the critical line

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