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Properties

Label 12-4030e6-1.1-c1e6-0-0
Degree $12$
Conductor $4.284\times 10^{21}$
Sign $1$
Analytic cond. $1.11043\times 10^{9}$
Root an. cond. $5.67271$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s − 3-s + 21·4-s − 6·5-s + 6·6-s + 4·7-s − 56·8-s − 9·9-s + 36·10-s − 6·11-s − 21·12-s + 6·13-s − 24·14-s + 6·15-s + 126·16-s − 4·17-s + 54·18-s + 3·19-s − 126·20-s − 4·21-s + 36·22-s − 7·23-s + 56·24-s + 21·25-s − 36·26-s + 10·27-s + 84·28-s + ⋯
L(s)  = 1  − 4.24·2-s − 0.577·3-s + 21/2·4-s − 2.68·5-s + 2.44·6-s + 1.51·7-s − 19.7·8-s − 3·9-s + 11.3·10-s − 1.80·11-s − 6.06·12-s + 1.66·13-s − 6.41·14-s + 1.54·15-s + 63/2·16-s − 0.970·17-s + 12.7·18-s + 0.688·19-s − 28.1·20-s − 0.872·21-s + 7.67·22-s − 1.45·23-s + 11.4·24-s + 21/5·25-s − 7.06·26-s + 1.92·27-s + 15.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6}\)
Sign: $1$
Analytic conductor: \(1.11043\times 10^{9}\)
Root analytic conductor: \(5.67271\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{6} \cdot 5^{6} \cdot 13^{6} \cdot 31^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
13 \( ( 1 - T )^{6} \)
31 \( ( 1 - T )^{6} \)
good3 \( 1 + T + 10 T^{2} + p^{2} T^{3} + 55 T^{4} + 46 T^{5} + 199 T^{6} + 46 p T^{7} + 55 p^{2} T^{8} + p^{5} T^{9} + 10 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 - 4 T + 32 T^{2} - 108 T^{3} + 501 T^{4} - 193 p T^{5} + 4483 T^{6} - 193 p^{2} T^{7} + 501 p^{2} T^{8} - 108 p^{3} T^{9} + 32 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 + 6 T + 67 T^{2} + 294 T^{3} + 1839 T^{4} + 6120 T^{5} + 26897 T^{6} + 6120 p T^{7} + 1839 p^{2} T^{8} + 294 p^{3} T^{9} + 67 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 + 4 T + 61 T^{2} + 214 T^{3} + 1960 T^{4} + 5989 T^{5} + 41577 T^{6} + 5989 p T^{7} + 1960 p^{2} T^{8} + 214 p^{3} T^{9} + 61 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 3 T + 89 T^{2} - 249 T^{3} + 3706 T^{4} - 8694 T^{5} + 90229 T^{6} - 8694 p T^{7} + 3706 p^{2} T^{8} - 249 p^{3} T^{9} + 89 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 7 T + 110 T^{2} + 695 T^{3} + 5642 T^{4} + 29287 T^{5} + 167495 T^{6} + 29287 p T^{7} + 5642 p^{2} T^{8} + 695 p^{3} T^{9} + 110 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 14 T + 175 T^{2} + 1346 T^{3} + 10696 T^{4} + 64679 T^{5} + 399897 T^{6} + 64679 p T^{7} + 10696 p^{2} T^{8} + 1346 p^{3} T^{9} + 175 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 14 T + 217 T^{2} - 1598 T^{3} + 13781 T^{4} - 68146 T^{5} + 517463 T^{6} - 68146 p T^{7} + 13781 p^{2} T^{8} - 1598 p^{3} T^{9} + 217 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 12 T + 221 T^{2} + 2059 T^{3} + 20319 T^{4} + 152202 T^{5} + 1061043 T^{6} + 152202 p T^{7} + 20319 p^{2} T^{8} + 2059 p^{3} T^{9} + 221 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 3 T + 161 T^{2} - 619 T^{3} + 11926 T^{4} - 50982 T^{5} + 589641 T^{6} - 50982 p T^{7} + 11926 p^{2} T^{8} - 619 p^{3} T^{9} + 161 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 2 T + 82 T^{2} + 332 T^{3} + 5597 T^{4} + 27503 T^{5} + 233059 T^{6} + 27503 p T^{7} + 5597 p^{2} T^{8} + 332 p^{3} T^{9} + 82 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 4 T + 193 T^{2} + 793 T^{3} + 17651 T^{4} + 77170 T^{5} + 1090885 T^{6} + 77170 p T^{7} + 17651 p^{2} T^{8} + 793 p^{3} T^{9} + 193 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 + 17 T + 436 T^{2} + 4955 T^{3} + 1190 p T^{4} + 578687 T^{5} + 5660475 T^{6} + 578687 p T^{7} + 1190 p^{3} T^{8} + 4955 p^{3} T^{9} + 436 p^{4} T^{10} + 17 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 5 T + 271 T^{2} + 1021 T^{3} + 33155 T^{4} + 93973 T^{5} + 2482049 T^{6} + 93973 p T^{7} + 33155 p^{2} T^{8} + 1021 p^{3} T^{9} + 271 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 8 T + 336 T^{2} - 2208 T^{3} + 50397 T^{4} - 267097 T^{5} + 4338835 T^{6} - 267097 p T^{7} + 50397 p^{2} T^{8} - 2208 p^{3} T^{9} + 336 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 16 T + 483 T^{2} + 5432 T^{3} + 89918 T^{4} + 750448 T^{5} + 8612943 T^{6} + 750448 p T^{7} + 89918 p^{2} T^{8} + 5432 p^{3} T^{9} + 483 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 - 11 T + 193 T^{2} - 845 T^{3} + 15875 T^{4} - 77401 T^{5} + 1531211 T^{6} - 77401 p T^{7} + 15875 p^{2} T^{8} - 845 p^{3} T^{9} + 193 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 2 T + 239 T^{2} - 162 T^{3} + 31983 T^{4} - 14056 T^{5} + 3192553 T^{6} - 14056 p T^{7} + 31983 p^{2} T^{8} - 162 p^{3} T^{9} + 239 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 4 T + 276 T^{2} - 2216 T^{3} + 37061 T^{4} - 379075 T^{5} + 3530313 T^{6} - 379075 p T^{7} + 37061 p^{2} T^{8} - 2216 p^{3} T^{9} + 276 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 + 8 T + 421 T^{2} + 2678 T^{3} + 80242 T^{4} + 408803 T^{5} + 9022551 T^{6} + 408803 p T^{7} + 80242 p^{2} T^{8} + 2678 p^{3} T^{9} + 421 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 5 T + 462 T^{2} - 1800 T^{3} + 96250 T^{4} - 295325 T^{5} + 11802805 T^{6} - 295325 p T^{7} + 96250 p^{2} T^{8} - 1800 p^{3} T^{9} + 462 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} \)
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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

−4.96409541967287766712365345501, −4.55021958983803807060937201893, −4.46265871228479865304577123906, −4.34205434424767594398605082917, −4.14585186490304313607349685528, −3.95098121000873719123485712736, −3.88485000808619510244029934497, −3.48917925516279355206434046052, −3.44013499898313449459210074525, −3.33013847459947770638616071071, −3.28170275695189749128659308309, −3.11483578690878701729939655385, −3.10856863040098657331200267781, −2.58712737449721497903583819479, −2.51980774280521954269543134845, −2.39492362632446232175485783124, −2.35136813010497921488819366933, −2.34100065934952280296335388819, −2.06044411436935169570414979318, −1.56706601868242486932187282827, −1.49354692480673237260072680139, −1.37839748941544264450739356015, −1.10903093294458217292974087119, −1.02428898621699656736610293968, −0.992673365017920534404414142596, 0, 0, 0, 0, 0, 0, 0.992673365017920534404414142596, 1.02428898621699656736610293968, 1.10903093294458217292974087119, 1.37839748941544264450739356015, 1.49354692480673237260072680139, 1.56706601868242486932187282827, 2.06044411436935169570414979318, 2.34100065934952280296335388819, 2.35136813010497921488819366933, 2.39492362632446232175485783124, 2.51980774280521954269543134845, 2.58712737449721497903583819479, 3.10856863040098657331200267781, 3.11483578690878701729939655385, 3.28170275695189749128659308309, 3.33013847459947770638616071071, 3.44013499898313449459210074525, 3.48917925516279355206434046052, 3.88485000808619510244029934497, 3.95098121000873719123485712736, 4.14585186490304313607349685528, 4.34205434424767594398605082917, 4.46265871228479865304577123906, 4.55021958983803807060937201893, 4.96409541967287766712365345501

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

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