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Properties

Label 12-1260e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.002\times 10^{18}$
Sign $1$
Analytic cond. $1.03725\times 10^{6}$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 7-s + 2·11-s − 10·13-s − 2·17-s − 19-s − 6·23-s + 3·25-s + 24·29-s + 3·31-s − 3·35-s + 9·37-s − 20·41-s − 2·43-s + 10·47-s − 49-s − 4·53-s + 6·55-s − 16·59-s − 2·61-s − 30·65-s + 5·67-s + 40·71-s + 15·73-s − 2·77-s + 13·79-s + 4·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s + 0.603·11-s − 2.77·13-s − 0.485·17-s − 0.229·19-s − 1.25·23-s + 3/5·25-s + 4.45·29-s + 0.538·31-s − 0.507·35-s + 1.47·37-s − 3.12·41-s − 0.304·43-s + 1.45·47-s − 1/7·49-s − 0.549·53-s + 0.809·55-s − 2.08·59-s − 0.256·61-s − 3.72·65-s + 0.610·67-s + 4.74·71-s + 1.75·73-s − 0.227·77-s + 1.46·79-s + 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(1.03725\times 10^{6}\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{12} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.231623077\)
\(L(\frac12)\) \(\approx\) \(4.231623077\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( ( 1 - T + T^{2} )^{3} \)
7 \( 1 + T + 2 T^{2} - 23 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
good11 \( 1 - 2 T - p T^{2} + 94 T^{3} - 64 T^{4} - 512 T^{5} + 3547 T^{6} - 512 p T^{7} - 64 p^{2} T^{8} + 94 p^{3} T^{9} - p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
13 \( ( 1 + 5 T + 28 T^{2} + 133 T^{3} + 28 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 + 2 T - 23 T^{2} - 118 T^{3} + 98 T^{4} + 1064 T^{5} + 3673 T^{6} + 1064 p T^{7} + 98 p^{2} T^{8} - 118 p^{3} T^{9} - 23 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + T - 39 T^{2} + 6 T^{3} + 823 T^{4} - 479 T^{5} - 16754 T^{6} - 479 p T^{7} + 823 p^{2} T^{8} + 6 p^{3} T^{9} - 39 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 6 T + 3 T^{2} - 30 T^{3} - 354 T^{4} - 2172 T^{5} - 6761 T^{6} - 2172 p T^{7} - 354 p^{2} T^{8} - 30 p^{3} T^{9} + 3 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
29 \( ( 1 - 12 T + 69 T^{2} - 318 T^{3} + 69 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{3} \)
37 \( 1 - 9 T - 9 T^{2} + 140 T^{3} + 495 T^{4} + 4869 T^{5} - 83658 T^{6} + 4869 p T^{7} + 495 p^{2} T^{8} + 140 p^{3} T^{9} - 9 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( ( 1 + 10 T + 45 T^{2} + 46 T^{3} + 45 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + T + 104 T^{2} + p T^{3} + 104 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 10 T - 5 T^{2} + 686 T^{3} - 2590 T^{4} - 14530 T^{5} + 221995 T^{6} - 14530 p T^{7} - 2590 p^{2} T^{8} + 686 p^{3} T^{9} - 5 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 4 T - 47 T^{2} - 884 T^{3} - 1642 T^{4} + 20308 T^{5} + 365077 T^{6} + 20308 p T^{7} - 1642 p^{2} T^{8} - 884 p^{3} T^{9} - 47 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 16 T + 13 T^{2} + 76 T^{3} + 14216 T^{4} + 60070 T^{5} - 331433 T^{6} + 60070 p T^{7} + 14216 p^{2} T^{8} + 76 p^{3} T^{9} + 13 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 2 T - 163 T^{2} - 162 T^{3} + 17206 T^{4} + 8110 T^{5} - 1200647 T^{6} + 8110 p T^{7} + 17206 p^{2} T^{8} - 162 p^{3} T^{9} - 163 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 5 T - 165 T^{2} + 396 T^{3} + 20005 T^{4} - 23015 T^{5} - 1476470 T^{6} - 23015 p T^{7} + 20005 p^{2} T^{8} + 396 p^{3} T^{9} - 165 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
71 \( ( 1 - 20 T + 327 T^{2} - 3038 T^{3} + 327 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 15 T - 21 T^{2} + 608 T^{3} + 10323 T^{4} - 69849 T^{5} - 220458 T^{6} - 69849 p T^{7} + 10323 p^{2} T^{8} + 608 p^{3} T^{9} - 21 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 13 T + 41 T^{2} - 462 T^{3} - 1541 T^{4} + 104851 T^{5} - 1001786 T^{6} + 104851 p T^{7} - 1541 p^{2} T^{8} - 462 p^{3} T^{9} + 41 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
83 \( ( 1 - 2 T + 93 T^{2} - 854 T^{3} + 93 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( 1 - 2 T - p T^{2} + 274 T^{3} - 88 T^{4} - 4124 T^{5} + 661393 T^{6} - 4124 p T^{7} - 88 p^{2} T^{8} + 274 p^{3} T^{9} - p^{5} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( ( 1 + 12 T + 267 T^{2} + 2212 T^{3} + 267 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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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

−5.07989105845013483136967306465, −4.97953937938728886460590077372, −4.81530529140326812234001336916, −4.58562788756823776599177066183, −4.57072633883612203697292191227, −4.51183673299178084779615880715, −4.11952161600281470905889368445, −3.95325330953352549180744078654, −3.94775411904096963295487524474, −3.58803828204228257449761781396, −3.25914691018663606359232890874, −3.24483418975072382144370981905, −3.19847423502928262805650442327, −2.76395632333989303178100066665, −2.66766835854641361369415402563, −2.46566065520123279098070208383, −2.31872018502216356391179474501, −2.26297750059148839374655435621, −1.91658902448448198267545590123, −1.81852406919958799205090834702, −1.58691421962066936039440818475, −1.15956370769686450564594056774, −0.791254406577318960248211619056, −0.72059499185396748098177922934, −0.30857671418426030382372427048, 0.30857671418426030382372427048, 0.72059499185396748098177922934, 0.791254406577318960248211619056, 1.15956370769686450564594056774, 1.58691421962066936039440818475, 1.81852406919958799205090834702, 1.91658902448448198267545590123, 2.26297750059148839374655435621, 2.31872018502216356391179474501, 2.46566065520123279098070208383, 2.66766835854641361369415402563, 2.76395632333989303178100066665, 3.19847423502928262805650442327, 3.24483418975072382144370981905, 3.25914691018663606359232890874, 3.58803828204228257449761781396, 3.94775411904096963295487524474, 3.95325330953352549180744078654, 4.11952161600281470905889368445, 4.51183673299178084779615880715, 4.57072633883612203697292191227, 4.58562788756823776599177066183, 4.81530529140326812234001336916, 4.97953937938728886460590077372, 5.07989105845013483136967306465

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

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