Properties

Label 12-4560e6-1.1-c1e6-0-1
Degree $12$
Conductor $8.991\times 10^{21}$
Sign $1$
Analytic cond. $2.33051\times 10^{9}$
Root an. cond. $6.03421$
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  − 6·3-s − 6·5-s + 21·9-s + 36·15-s − 4·17-s + 2·19-s + 21·25-s − 56·27-s − 12·31-s − 126·45-s + 22·49-s + 24·51-s − 12·57-s + 8·59-s + 4·67-s + 8·71-s − 126·75-s + 40·79-s + 126·81-s + 24·85-s + 72·93-s − 12·95-s + 12·101-s + 16·103-s + 28·121-s − 56·125-s + 127-s + ⋯
L(s)  = 1  − 3.46·3-s − 2.68·5-s + 7·9-s + 9.29·15-s − 0.970·17-s + 0.458·19-s + 21/5·25-s − 10.7·27-s − 2.15·31-s − 18.7·45-s + 22/7·49-s + 3.36·51-s − 1.58·57-s + 1.04·59-s + 0.488·67-s + 0.949·71-s − 14.5·75-s + 4.50·79-s + 14·81-s + 2.60·85-s + 7.46·93-s − 1.23·95-s + 1.19·101-s + 1.57·103-s + 2.54·121-s − 5.00·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\)
Sign: $1$
Analytic conductor: \(2.33051\times 10^{9}\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.039242761\)
\(L(\frac12)\) \(\approx\) \(1.039242761\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T )^{6} \)
5 \( ( 1 + T )^{6} \)
19 \( 1 - 2 T - 3 T^{2} - 52 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
good7 \( 1 - 22 T^{2} + 293 T^{4} - 2440 T^{6} + 293 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 28 T^{2} + 285 T^{4} - 2048 T^{6} + 285 p^{2} T^{8} - 28 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 - 58 T^{2} + 1613 T^{4} - 26536 T^{6} + 1613 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} \)
17 \( ( 1 + 2 T + 37 T^{2} + 56 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 40 T^{2} + 1095 T^{4} - 29312 T^{6} + 1095 p^{2} T^{8} - 40 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 - 136 T^{2} + 8349 T^{4} - 304160 T^{6} + 8349 p^{2} T^{8} - 136 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 + 6 T + 27 T^{2} - 32 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 130 T^{2} + 7661 T^{4} - 312232 T^{6} + 7661 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} \)
41 \( 1 - 192 T^{2} + 16629 T^{4} - 855808 T^{6} + 16629 p^{2} T^{8} - 192 p^{4} T^{10} + p^{6} T^{12} \)
43 \( 1 - 166 T^{2} + 12653 T^{4} - 633256 T^{6} + 12653 p^{2} T^{8} - 166 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 136 T^{2} + 10359 T^{4} - 579776 T^{6} + 10359 p^{2} T^{8} - 136 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 220 T^{2} + 23535 T^{4} - 1553432 T^{6} + 23535 p^{2} T^{8} - 220 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 4 T + 121 T^{2} - 376 T^{3} + 121 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 + 105 T^{2} - 256 T^{3} + 105 p T^{4} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - 2 T + 105 T^{2} - 172 T^{3} + 105 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 4 T + 157 T^{2} - 472 T^{3} + 157 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 + 15 T^{2} - 976 T^{3} + 15 p T^{4} + p^{3} T^{6} )^{2} \)
79 \( ( 1 - 20 T + 281 T^{2} - 3032 T^{3} + 281 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 336 T^{2} + 56463 T^{4} - 5802352 T^{6} + 56463 p^{2} T^{8} - 336 p^{4} T^{10} + p^{6} T^{12} \)
89 \( 1 - 336 T^{2} + 55077 T^{4} - 5833600 T^{6} + 55077 p^{2} T^{8} - 336 p^{4} T^{10} + p^{6} T^{12} \)
97 \( 1 + 30 T^{2} + 7413 T^{4} + 1025944 T^{6} + 7413 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.32795370068358497285731687450, −4.12434096134390871778137292017, −4.10401730123494150969906358395, −3.90530412683996598864837199260, −3.81107837955976643731272972020, −3.61048863443863980418941186053, −3.58314662754336707994178878673, −3.46586520692010833332600848957, −3.25149379641849326733817922861, −3.05746133603291805649396212742, −2.92230921525461363192249061481, −2.73542969040162127143386050123, −2.37772955929472110090827996593, −2.23215539739749270541178352634, −2.16824617437862267716294089307, −1.91941063877319390263066336520, −1.79373406968755348077345809360, −1.62269645581423776388055752132, −1.42754528244400970001071374784, −0.926937810301434658420741285502, −0.76694376556216454007553130977, −0.70632541317693189676312896758, −0.66013527500673227512376064884, −0.42004704081597296589588180401, −0.28993997801605106580078122689, 0.28993997801605106580078122689, 0.42004704081597296589588180401, 0.66013527500673227512376064884, 0.70632541317693189676312896758, 0.76694376556216454007553130977, 0.926937810301434658420741285502, 1.42754528244400970001071374784, 1.62269645581423776388055752132, 1.79373406968755348077345809360, 1.91941063877319390263066336520, 2.16824617437862267716294089307, 2.23215539739749270541178352634, 2.37772955929472110090827996593, 2.73542969040162127143386050123, 2.92230921525461363192249061481, 3.05746133603291805649396212742, 3.25149379641849326733817922861, 3.46586520692010833332600848957, 3.58314662754336707994178878673, 3.61048863443863980418941186053, 3.81107837955976643731272972020, 3.90530412683996598864837199260, 4.10401730123494150969906358395, 4.12434096134390871778137292017, 4.32795370068358497285731687450

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

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