Properties

Label 12-825e6-1.1-c3e6-0-0
Degree $12$
Conductor $3.153\times 10^{17}$
Sign $1$
Analytic cond. $1.33020\times 10^{10}$
Root an. cond. $6.97686$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 27·9-s + 66·11-s − 61·16-s − 292·19-s − 136·29-s − 136·31-s − 54·36-s − 392·41-s + 132·44-s + 1.18e3·49-s + 2.08e3·59-s + 1.28e3·61-s − 380·64-s − 1.08e3·71-s − 584·76-s + 3.17e3·79-s + 486·81-s + 4.24e3·89-s − 1.78e3·99-s − 144·101-s − 1.20e3·109-s − 272·116-s + 2.54e3·121-s − 272·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1/4·4-s − 9-s + 1.80·11-s − 0.953·16-s − 3.52·19-s − 0.870·29-s − 0.787·31-s − 1/4·36-s − 1.49·41-s + 0.452·44-s + 3.45·49-s + 4.60·59-s + 2.69·61-s − 0.742·64-s − 1.81·71-s − 0.881·76-s + 4.51·79-s + 2/3·81-s + 5.05·89-s − 1.80·99-s − 0.141·101-s − 1.05·109-s − 0.217·116-s + 1.90·121-s − 0.196·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{12} \cdot 11^{6}\)
Sign: $1$
Analytic conductor: \(1.33020\times 10^{10}\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{12} \cdot 11^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(1.546406890\)
\(L(\frac12)\) \(\approx\) \(1.546406890\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{2} T^{2} )^{3} \)
5 \( 1 \)
11 \( ( 1 - p T )^{6} \)
good2 \( 1 - p T^{2} + 65 T^{4} + p^{7} T^{6} + 65 p^{6} T^{8} - p^{13} T^{10} + p^{18} T^{12} \)
7 \( 1 - 1186 T^{2} + 111641 p T^{4} - 323726332 T^{6} + 111641 p^{7} T^{8} - 1186 p^{12} T^{10} + p^{18} T^{12} \)
13 \( 1 - 6062 T^{2} + 22505175 T^{4} - 60410692644 T^{6} + 22505175 p^{6} T^{8} - 6062 p^{12} T^{10} + p^{18} T^{12} \)
17 \( 1 - 1314 T^{2} + 281903 p T^{4} - 165557971772 T^{6} + 281903 p^{7} T^{8} - 1314 p^{12} T^{10} + p^{18} T^{12} \)
19 \( ( 1 + 146 T + 25953 T^{2} + 2033788 T^{3} + 25953 p^{3} T^{4} + 146 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
23 \( 1 - 27658 T^{2} + 20210527 T^{4} + 4094647717876 T^{6} + 20210527 p^{6} T^{8} - 27658 p^{12} T^{10} + p^{18} T^{12} \)
29 \( ( 1 + 68 T + 18803 T^{2} + 153848 T^{3} + 18803 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
31 \( ( 1 + 68 T + 66141 T^{2} + 2239480 T^{3} + 66141 p^{3} T^{4} + 68 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
37 \( 1 - 6074 p T^{2} + 23289627015 T^{4} - 1465686888981756 T^{6} + 23289627015 p^{6} T^{8} - 6074 p^{13} T^{10} + p^{18} T^{12} \)
41 \( ( 1 + 196 T + 4407 p T^{2} + 22652824 T^{3} + 4407 p^{4} T^{4} + 196 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
43 \( 1 - 145130 T^{2} + 22860058583 T^{4} - 1827357668621708 T^{6} + 22860058583 p^{6} T^{8} - 145130 p^{12} T^{10} + p^{18} T^{12} \)
47 \( 1 - 347498 T^{2} + 68290484975 T^{4} - 8487381369946124 T^{6} + 68290484975 p^{6} T^{8} - 347498 p^{12} T^{10} + p^{18} T^{12} \)
53 \( 1 - 346594 T^{2} + 62458330663 T^{4} - 8266498983743612 T^{6} + 62458330663 p^{6} T^{8} - 346594 p^{12} T^{10} + p^{18} T^{12} \)
59 \( ( 1 - 1044 T + 680281 T^{2} - 344604088 T^{3} + 680281 p^{3} T^{4} - 1044 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
61 \( ( 1 - 642 T + 620395 T^{2} - 268686220 T^{3} + 620395 p^{3} T^{4} - 642 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
67 \( 1 - 824466 T^{2} + 350068488567 T^{4} - 108165802801072732 T^{6} + 350068488567 p^{6} T^{8} - 824466 p^{12} T^{10} + p^{18} T^{12} \)
71 \( ( 1 + 544 T + 944005 T^{2} + 395960768 T^{3} + 944005 p^{3} T^{4} + 544 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
73 \( 1 - 924950 T^{2} + 156926020575 T^{4} + 40147697043546060 T^{6} + 156926020575 p^{6} T^{8} - 924950 p^{12} T^{10} + p^{18} T^{12} \)
79 \( ( 1 - 1586 T + 2041325 T^{2} - 1549224716 T^{3} + 2041325 p^{3} T^{4} - 1586 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
83 \( 1 - 312710 T^{2} + 792973516471 T^{4} - 221260272194940724 T^{6} + 792973516471 p^{6} T^{8} - 312710 p^{12} T^{10} + p^{18} T^{12} \)
89 \( ( 1 - 2122 T + 3521847 T^{2} - 3285333068 T^{3} + 3521847 p^{3} T^{4} - 2122 p^{6} T^{5} + p^{9} T^{6} )^{2} \)
97 \( 1 - 2510714 T^{2} + 3061284302031 T^{4} - 2875684422212751852 T^{6} + 3061284302031 p^{6} T^{8} - 2510714 p^{12} T^{10} + p^{18} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.93211562044559100897948968175, −4.83794198636038342370295609915, −4.63157648317143216868314167908, −4.58685377708898067915749473034, −4.20726825036438121215678725716, −4.09913153835463025329366251626, −3.94590424797836244282359280031, −3.85323246458169940572605926418, −3.66750897630696223020547677680, −3.50637291121786958191370652154, −3.39635407773799364715554303795, −3.14235106942633049239877996582, −2.90332956695719562614345734434, −2.44245855285065307557771572600, −2.17489588850634757249021188060, −2.13871012016743284835804207660, −2.10591576118162003286585364675, −2.09175622440596685539037590006, −2.07141164971225816415223993932, −1.30572348185022477284402139914, −0.920839449375790964590036914452, −0.907785664779031561846166386016, −0.880889683012004442283545164812, −0.29333210226067697247864627490, −0.14572755111208684153193827265, 0.14572755111208684153193827265, 0.29333210226067697247864627490, 0.880889683012004442283545164812, 0.907785664779031561846166386016, 0.920839449375790964590036914452, 1.30572348185022477284402139914, 2.07141164971225816415223993932, 2.09175622440596685539037590006, 2.10591576118162003286585364675, 2.13871012016743284835804207660, 2.17489588850634757249021188060, 2.44245855285065307557771572600, 2.90332956695719562614345734434, 3.14235106942633049239877996582, 3.39635407773799364715554303795, 3.50637291121786958191370652154, 3.66750897630696223020547677680, 3.85323246458169940572605926418, 3.94590424797836244282359280031, 4.09913153835463025329366251626, 4.20726825036438121215678725716, 4.58685377708898067915749473034, 4.63157648317143216868314167908, 4.83794198636038342370295609915, 4.93211562044559100897948968175

Graph of the $Z$-function along the critical line

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