Properties

Label 12-1680e6-1.1-c1e6-0-4
Degree $12$
Conductor $2.248\times 10^{19}$
Sign $1$
Analytic cond. $5.82798\times 10^{6}$
Root an. cond. $3.66263$
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  − 2·5-s − 3·9-s − 4·11-s + 4·19-s + 25-s − 4·29-s + 4·31-s + 28·41-s + 6·45-s − 3·49-s + 8·55-s + 16·59-s + 4·61-s + 12·71-s − 40·79-s + 6·81-s − 4·89-s − 8·95-s + 12·99-s + 60·101-s + 20·109-s − 14·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s − 1.20·11-s + 0.917·19-s + 1/5·25-s − 0.742·29-s + 0.718·31-s + 4.37·41-s + 0.894·45-s − 3/7·49-s + 1.07·55-s + 2.08·59-s + 0.512·61-s + 1.42·71-s − 4.50·79-s + 2/3·81-s − 0.423·89-s − 0.820·95-s + 1.20·99-s + 5.97·101-s + 1.91·109-s − 1.27·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(5.82798\times 10^{6}\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.552931634\)
\(L(\frac12)\) \(\approx\) \(5.552931634\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T^{2} )^{3} \)
5 \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( ( 1 + T^{2} )^{3} \)
good11 \( ( 1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( ( 1 - 8 T + 7 T^{2} + 64 T^{3} + 7 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )( 1 + 8 T + 7 T^{2} - 64 T^{3} + 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} ) \)
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 \( ( 1 - 2 T + 45 T^{2} - 68 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 - 90 T^{2} + 3839 T^{4} - 105452 T^{6} + 3839 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
31 \( ( 1 - 2 T + 81 T^{2} - 116 T^{3} + 81 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 10 T - 5 T^{2} + 356 T^{3} - 5 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )( 1 + 10 T - 5 T^{2} - 356 T^{3} - 5 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} ) \)
41 \( ( 1 - 14 T + 175 T^{2} - 1188 T^{3} + 175 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 82 T^{2} + 5399 T^{4} - 217692 T^{6} + 5399 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \)
47 \( 1 - 90 T^{2} + 5231 T^{4} - 236204 T^{6} + 5231 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \)
53 \( 1 - 66 T^{2} + 2711 T^{4} + 8452 T^{6} + 2711 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 - 8 T + 113 T^{2} - 688 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( ( 1 - p T^{2} )^{6} \)
71 \( ( 1 - 6 T + 113 T^{2} - 1052 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 154 T^{2} + 21503 T^{4} - 1698540 T^{6} + 21503 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 20 T + 317 T^{2} + 3096 T^{3} + 317 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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 \( ( 1 + 2 T + 207 T^{2} + 156 T^{3} + 207 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 394 T^{2} + 77839 T^{4} - 9393484 T^{6} + 77839 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} \)
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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

−5.03956177798313009773268344023, −4.57948093329126913116799050015, −4.53436903673034710251402601475, −4.41724729922782402160497539854, −4.38688981097203963080989528665, −4.07192536679267238635122808977, −4.04591397487981615724599830567, −3.79210960496064903669336808824, −3.53795236370793453551078348072, −3.50381520205702350857196039088, −3.30532378375323121051098202255, −3.13613461209930975958805121364, −2.78684825010035653851783874512, −2.75027625346384550231324048307, −2.73013217538304145810316858041, −2.50126429883998132797403839887, −2.36692912680192085570046013664, −1.89321449668846189195373127875, −1.86846097769975374467552395638, −1.66821510282946547914017341765, −1.44948508828435448526484678577, −0.76941064174867645652252405683, −0.68641881922207452473164270519, −0.55617799536496794240769176047, −0.53172716341158858796369340894, 0.53172716341158858796369340894, 0.55617799536496794240769176047, 0.68641881922207452473164270519, 0.76941064174867645652252405683, 1.44948508828435448526484678577, 1.66821510282946547914017341765, 1.86846097769975374467552395638, 1.89321449668846189195373127875, 2.36692912680192085570046013664, 2.50126429883998132797403839887, 2.73013217538304145810316858041, 2.75027625346384550231324048307, 2.78684825010035653851783874512, 3.13613461209930975958805121364, 3.30532378375323121051098202255, 3.50381520205702350857196039088, 3.53795236370793453551078348072, 3.79210960496064903669336808824, 4.04591397487981615724599830567, 4.07192536679267238635122808977, 4.38688981097203963080989528665, 4.41724729922782402160497539854, 4.53436903673034710251402601475, 4.57948093329126913116799050015, 5.03956177798313009773268344023

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

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