Properties

Degree 12
Conductor $ 2^{30} \cdot 3^{12} \cdot 5^{6} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·13-s + 25-s + 16·31-s − 16·37-s + 4·41-s + 18·49-s − 24·53-s + 16·71-s − 16·79-s − 16·83-s + 20·89-s − 24·107-s + 34·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.21·13-s + 1/5·25-s + 2.87·31-s − 2.63·37-s + 0.624·41-s + 18/7·49-s − 3.29·53-s + 1.89·71-s − 1.80·79-s − 1.75·83-s + 2.11·89-s − 2.32·107-s + 3.09·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(2^{30} \cdot 3^{12} \cdot 5^{6}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1440} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 2^{30} \cdot 3^{12} \cdot 5^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(2.937563927\)
\(L(\frac12)\)  \(\approx\)  \(2.937563927\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T^{2} + 8 T^{3} - p T^{4} + p^{3} T^{6} \)
good7 \( 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 4 T + 23 T^{2} + 48 T^{3} + 23 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
17 \( 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
29 \( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 - 8 T + 89 T^{2} - 432 T^{3} + 89 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( ( 1 - 2 T + 23 T^{2} - 220 T^{3} + 23 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 + 65 T^{2} - 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2} \)
47 \( 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 12 T + 191 T^{2} + 1264 T^{3} + 191 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 + 137 T^{2} + 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2} \)
71 \( ( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 233 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 8 T + 185 T^{2} + 880 T^{3} + 185 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−4.93971405099376057097786525951, −4.90084037144086778086310483785, −4.83587466011782058375013325307, −4.48742785455651352512099117769, −4.39829816710960920915233857087, −4.27779972281684461919115233420, −4.14661494412211771573830890763, −3.88653999397147373953762516926, −3.72292975705491023808755920755, −3.58491394484793003804569950634, −3.35456174967073238826948458896, −3.04590940349524161916185500591, −2.95466635946635248793854288529, −2.88391517053527830766255718598, −2.77057768298575599662989351449, −2.34532674122059205921360451456, −2.30279813199603973688039819981, −2.29565540139803895615350775928, −1.69220754793623285507704848024, −1.68848372232822419363629281888, −1.55838631837197677616013559892, −1.20762817364506137780143491830, −0.65635533558555377851112369739, −0.63387276770046806625883227740, −0.29588614584358726466541293078, 0.29588614584358726466541293078, 0.63387276770046806625883227740, 0.65635533558555377851112369739, 1.20762817364506137780143491830, 1.55838631837197677616013559892, 1.68848372232822419363629281888, 1.69220754793623285507704848024, 2.29565540139803895615350775928, 2.30279813199603973688039819981, 2.34532674122059205921360451456, 2.77057768298575599662989351449, 2.88391517053527830766255718598, 2.95466635946635248793854288529, 3.04590940349524161916185500591, 3.35456174967073238826948458896, 3.58491394484793003804569950634, 3.72292975705491023808755920755, 3.88653999397147373953762516926, 4.14661494412211771573830890763, 4.27779972281684461919115233420, 4.39829816710960920915233857087, 4.48742785455651352512099117769, 4.83587466011782058375013325307, 4.90084037144086778086310483785, 4.93971405099376057097786525951

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

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