Properties

Label 12-1323e6-1.1-c1e6-0-5
Degree $12$
Conductor $5.362\times 10^{18}$
Sign $1$
Analytic cond. $1.39002\times 10^{6}$
Root an. cond. $3.25026$
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  − 2-s + 2·4-s + 10·5-s + 8-s − 10·10-s + 4·11-s − 3·13-s − 12·17-s + 3·19-s + 20·20-s − 4·22-s + 41·25-s + 3·26-s + 29-s + 3·31-s + 4·32-s + 12·34-s + 3·37-s − 3·38-s + 10·40-s − 22·41-s + 3·43-s + 8·44-s − 9·47-s − 41·50-s − 6·52-s − 18·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 4-s + 4.47·5-s + 0.353·8-s − 3.16·10-s + 1.20·11-s − 0.832·13-s − 2.91·17-s + 0.688·19-s + 4.47·20-s − 0.852·22-s + 41/5·25-s + 0.588·26-s + 0.185·29-s + 0.538·31-s + 0.707·32-s + 2.05·34-s + 0.493·37-s − 0.486·38-s + 1.58·40-s − 3.43·41-s + 0.457·43-s + 1.20·44-s − 1.31·47-s − 5.79·50-s − 0.832·52-s − 2.47·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{18} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{18} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.39002\times 10^{6}\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{18} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.957358905\)
\(L(\frac12)\) \(\approx\) \(9.957358905\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + T - T^{2} - p^{2} T^{3} - 3 T^{4} + p T^{5} + 13 T^{6} + p^{2} T^{7} - 3 p^{2} T^{8} - p^{5} T^{9} - p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
5 \( ( 1 - p T + 17 T^{2} - 39 T^{3} + 17 p T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2} \)
11 \( ( 1 - 2 T + 14 T^{2} + 3 T^{3} + 14 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{3} \)
17 \( 1 + 12 T + 54 T^{2} + 210 T^{3} + 1350 T^{4} + 5898 T^{5} + 19735 T^{6} + 5898 p T^{7} + 1350 p^{2} T^{8} + 210 p^{3} T^{9} + 54 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T - 42 T^{2} + 61 T^{3} + 69 p T^{4} - 726 T^{5} - 27501 T^{6} - 726 p T^{7} + 69 p^{3} T^{8} + 61 p^{3} T^{9} - 42 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( ( 1 + 36 T^{2} + 9 T^{3} + 36 p T^{4} + p^{3} T^{6} )^{2} \)
29 \( 1 - T - 82 T^{2} + 31 T^{3} + 4425 T^{4} - 758 T^{5} - 148595 T^{6} - 758 p T^{7} + 4425 p^{2} T^{8} + 31 p^{3} T^{9} - 82 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 3 T - 60 T^{2} + 219 T^{3} + 1983 T^{4} - 4746 T^{5} - 51289 T^{6} - 4746 p T^{7} + 1983 p^{2} T^{8} + 219 p^{3} T^{9} - 60 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 3 T - 48 T^{2} + 435 T^{3} + 231 T^{4} - 8724 T^{5} + 60581 T^{6} - 8724 p T^{7} + 231 p^{2} T^{8} + 435 p^{3} T^{9} - 48 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 22 T + 206 T^{2} + 1802 T^{3} + 18432 T^{4} + 135116 T^{5} + 808243 T^{6} + 135116 p T^{7} + 18432 p^{2} T^{8} + 1802 p^{3} T^{9} + 206 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T - 54 T^{2} + 569 T^{3} + 123 T^{4} - 13170 T^{5} + 115347 T^{6} - 13170 p T^{7} + 123 p^{2} T^{8} + 569 p^{3} T^{9} - 54 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 9 T - 6 T^{2} - 531 T^{3} - 2433 T^{4} + 3438 T^{5} + 104623 T^{6} + 3438 p T^{7} - 2433 p^{2} T^{8} - 531 p^{3} T^{9} - 6 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 18 T + 90 T^{2} + 378 T^{3} + 7848 T^{4} + 52668 T^{5} + 160459 T^{6} + 52668 p T^{7} + 7848 p^{2} T^{8} + 378 p^{3} T^{9} + 90 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 9 T - 90 T^{2} - 459 T^{3} + 10161 T^{4} + 20556 T^{5} - 598421 T^{6} + 20556 p T^{7} + 10161 p^{2} T^{8} - 459 p^{3} T^{9} - 90 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 6 T - 126 T^{2} + 358 T^{3} + 12372 T^{4} - 11472 T^{5} - 838653 T^{6} - 11472 p T^{7} + 12372 p^{2} T^{8} + 358 p^{3} T^{9} - 126 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 6 T^{2} + 1366 T^{3} + 438 T^{4} + 4098 T^{5} + 1065603 T^{6} + 4098 p T^{7} + 438 p^{2} T^{8} + 1366 p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 9 T + 207 T^{2} + 1197 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 3 T - 42 T^{2} - 1209 T^{3} - 3165 T^{4} + 28380 T^{5} + 1003961 T^{6} + 28380 p T^{7} - 3165 p^{2} T^{8} - 1209 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 15 T + 36 T^{2} - 367 T^{3} - 3225 T^{4} - 51726 T^{5} - 676905 T^{6} - 51726 p T^{7} - 3225 p^{2} T^{8} - 367 p^{3} T^{9} + 36 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 + 12 T - 144 T^{2} - 582 T^{3} + 34812 T^{4} + 90444 T^{5} - 2656433 T^{6} + 90444 p T^{7} + 34812 p^{2} T^{8} - 582 p^{3} T^{9} - 144 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 2 T - 112 T^{2} - 1238 T^{3} + 1662 T^{4} + 59806 T^{5} + 720895 T^{6} + 59806 p T^{7} + 1662 p^{2} T^{8} - 1238 p^{3} T^{9} - 112 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 3 T - 168 T^{2} + 573 T^{3} + 14223 T^{4} - 78504 T^{5} - 1297807 T^{6} - 78504 p T^{7} + 14223 p^{2} T^{8} + 573 p^{3} T^{9} - 168 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \)
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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

−5.11992526883667907070194263685, −5.01947013554723724510571920205, −4.88527795041514598952790044638, −4.62158125647881367813605239187, −4.54450833531746815685850715548, −4.20996601225204761944537345202, −4.20231391995656008306432326709, −4.05400179613166386291722245731, −3.90620740469984902762392093388, −3.45475130251187506488412073808, −3.23169038142805804764879053256, −3.03647093777611315681052334090, −2.93286689312423162309713108111, −2.79662716044767984207115529881, −2.54597394948905562684377700139, −2.53094977480918447898123170528, −2.05049902883530068989434910254, −1.88828143425112024942326679699, −1.88031270194602658412712590091, −1.63683742336327449541235780238, −1.57043525909738137125339212538, −1.46443101212424253901560924938, −1.42787887609306606369133999650, −0.53497908559497811774259293031, −0.40322055123403983971888824935, 0.40322055123403983971888824935, 0.53497908559497811774259293031, 1.42787887609306606369133999650, 1.46443101212424253901560924938, 1.57043525909738137125339212538, 1.63683742336327449541235780238, 1.88031270194602658412712590091, 1.88828143425112024942326679699, 2.05049902883530068989434910254, 2.53094977480918447898123170528, 2.54597394948905562684377700139, 2.79662716044767984207115529881, 2.93286689312423162309713108111, 3.03647093777611315681052334090, 3.23169038142805804764879053256, 3.45475130251187506488412073808, 3.90620740469984902762392093388, 4.05400179613166386291722245731, 4.20231391995656008306432326709, 4.20996601225204761944537345202, 4.54450833531746815685850715548, 4.62158125647881367813605239187, 4.88527795041514598952790044638, 5.01947013554723724510571920205, 5.11992526883667907070194263685

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

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