Properties

Label 12-2205e6-1.1-c1e6-0-0
Degree $12$
Conductor $1.149\times 10^{20}$
Sign $1$
Analytic cond. $2.97929\times 10^{7}$
Root an. cond. $4.19607$
Motivic weight $1$
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  + 4-s + 2·5-s − 12·11-s + 2·16-s + 12·19-s + 2·20-s + 25-s + 4·29-s − 4·31-s + 4·41-s − 12·44-s − 24·55-s − 32·59-s + 12·61-s + 6·64-s − 12·71-s + 12·76-s − 24·79-s + 4·80-s − 28·89-s + 24·95-s + 100-s − 44·101-s + 20·109-s + 4·116-s + 18·121-s − 4·124-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s − 3.61·11-s + 1/2·16-s + 2.75·19-s + 0.447·20-s + 1/5·25-s + 0.742·29-s − 0.718·31-s + 0.624·41-s − 1.80·44-s − 3.23·55-s − 4.16·59-s + 1.53·61-s + 3/4·64-s − 1.42·71-s + 1.37·76-s − 2.70·79-s + 0.447·80-s − 2.96·89-s + 2.46·95-s + 1/10·100-s − 4.37·101-s + 1.91·109-s + 0.371·116-s + 1.63·121-s − 0.359·124-s + ⋯

Functional equation

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

Invariants

Degree: \(12\)
Conductor: \(3^{12} \cdot 5^{6} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(2.97929\times 10^{7}\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2205} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{12} \cdot 5^{6} \cdot 7^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2350296301\)
\(L(\frac12)\) \(\approx\) \(0.2350296301\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( 1 \)
good2 \( 1 - T^{2} - T^{4} - 3 T^{6} - p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
11 \( ( 1 + 2 T + p T^{2} )^{6} \)
13 \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 106 T^{2} + 5183 T^{4} - 150348 T^{6} + 5183 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 + 2 T + 41 T^{2} - 60 T^{3} + 41 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 46 T^{2} + 1399 T^{4} - 74788 T^{6} + 1399 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 154 T^{2} + 12143 T^{4} - 652332 T^{6} + 12143 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 146 T^{2} + 14103 T^{4} - 884828 T^{6} + 14103 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 16 T + 113 T^{2} + 608 T^{3} + 113 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 6 T + 131 T^{2} - 484 T^{3} + 131 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 + 2 T + p T^{2} )^{6} \)
73 \( 1 - 298 T^{2} + 43775 T^{4} - 3982284 T^{6} + 43775 p^{2} T^{8} - 298 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 12 T + 221 T^{2} + 1576 T^{3} + 221 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 + 14 T + 319 T^{2} + 2532 T^{3} + 319 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.99068136947622664915973289956, −4.35342739496451027615910363289, −4.33126856157953412719783441945, −4.32080834356785249897568255854, −4.14468843510134093452987759364, −4.12444642450554424963957119664, −3.99056743567381963463129263038, −3.31620368900512514513234200802, −3.26489668343395768803364311984, −3.10936214990072829714542304890, −3.07517751012712589630040943451, −3.05207443399162592910040032565, −2.96411714493426832030478141394, −2.65680067210576098649445646880, −2.56547696483063511763760074006, −2.40785019656746041281598322326, −2.03097240904092881299620156597, −1.82655398677392774121832209764, −1.74018171159303724766599083988, −1.69731906809789943965640770375, −1.22215213643024866436128561551, −1.18837237314975536193141472076, −0.73423012105103947301503000113, −0.56550848713064643420962290903, −0.05487210510005795606610631152, 0.05487210510005795606610631152, 0.56550848713064643420962290903, 0.73423012105103947301503000113, 1.18837237314975536193141472076, 1.22215213643024866436128561551, 1.69731906809789943965640770375, 1.74018171159303724766599083988, 1.82655398677392774121832209764, 2.03097240904092881299620156597, 2.40785019656746041281598322326, 2.56547696483063511763760074006, 2.65680067210576098649445646880, 2.96411714493426832030478141394, 3.05207443399162592910040032565, 3.07517751012712589630040943451, 3.10936214990072829714542304890, 3.26489668343395768803364311984, 3.31620368900512514513234200802, 3.99056743567381963463129263038, 4.12444642450554424963957119664, 4.14468843510134093452987759364, 4.32080834356785249897568255854, 4.33126856157953412719783441945, 4.35342739496451027615910363289, 4.99068136947622664915973289956

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

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