Properties

Degree 24
Conductor $ 2^{72} \cdot 3^{24} \cdot 7^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·25-s − 8·37-s − 6·49-s − 56·61-s + 80·109-s − 76·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  + 24/5·25-s − 1.31·37-s − 6/7·49-s − 7.17·61-s + 7.66·109-s − 6.90·121-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 − 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{72} \cdot 3^{24} \cdot 7^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4032} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 2^{72} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $0.02893216511$
$L(\frac12)$  $\approx$  $0.02893216511$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( ( 1 + T^{2} )^{6} \)
good5 \( ( 1 - 12 T^{2} + 19 p T^{4} - 568 T^{6} + 19 p^{3} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 + 38 T^{2} + 65 p T^{4} + 9068 T^{6} + 65 p^{3} T^{8} + 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 + 11 T^{2} + 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{4} \)
17 \( ( 1 - 4 p T^{2} + 2167 T^{4} - 44168 T^{6} + 2167 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 - 50 T^{2} + 1639 T^{4} - 35804 T^{6} + 1639 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
23 \( ( 1 - 2 T^{2} + 979 T^{4} + 164 p T^{6} + 979 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{6} \)
31 \( ( 1 - 58 T^{2} + 3727 T^{4} - 113644 T^{6} + 3727 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 + 2 T + 47 T^{2} + 276 T^{3} + 47 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
41 \( ( 1 - 132 T^{2} + 10823 T^{4} - 12872 p T^{6} + 10823 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 - 98 T^{2} + 4567 T^{4} - 172988 T^{6} + 4567 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 10 T^{2} + 4591 T^{4} + 70732 T^{6} + 4591 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 184 T^{2} + 16171 T^{4} - 975856 T^{6} + 16171 p^{2} T^{8} - 184 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 + 146 T^{2} + 7799 T^{4} + 306396 T^{6} + 7799 p^{2} T^{8} + 146 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 14 T + 187 T^{2} + 1700 T^{3} + 187 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
67 \( ( 1 - 186 T^{2} + 13175 T^{4} - 680684 T^{6} + 13175 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 + 158 T^{2} + 5683 T^{4} - 107140 T^{6} + 5683 p^{2} T^{8} + 158 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 + 47 T^{2} - 352 T^{3} + 47 p T^{4} + p^{3} T^{6} )^{4} \)
79 \( ( 1 - 162 T^{2} + 19359 T^{4} - 2006332 T^{6} + 19359 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( ( 1 + 370 T^{2} + 65191 T^{4} + 6834652 T^{6} + 65191 p^{2} T^{8} + 370 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 308 T^{2} + 49895 T^{4} - 5241384 T^{6} + 49895 p^{2} T^{8} - 308 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 + 135 T^{2} + 128 T^{3} + 135 p T^{4} + p^{3} T^{6} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.60200613790308956010116932702, −2.47977778246253514303772233892, −2.37971237986761298207558292979, −2.19296232636556929616015510186, −2.17784153476863363437713306591, −2.14326820126507998570725508745, −2.09748041020420391754801955811, −1.98933741999365350588236977846, −1.76824760168611945423313129837, −1.74357723956476554793062401648, −1.66299146394683237871553114404, −1.54286010113212644936170196913, −1.41086467593643744262265600826, −1.36891454208636501227539696840, −1.23606982104351875331330739030, −1.19015302654336680262086316331, −1.07532066234199239964883932544, −1.04866839736139433784435951726, −0.984519622445090559170399204414, −0.875278314986834915823927579903, −0.60285681886471432230095864836, −0.50113396288982743828237085438, −0.28775773210053779626743692482, −0.11235887295961448298586996020, −0.01972896319906616089865538733, 0.01972896319906616089865538733, 0.11235887295961448298586996020, 0.28775773210053779626743692482, 0.50113396288982743828237085438, 0.60285681886471432230095864836, 0.875278314986834915823927579903, 0.984519622445090559170399204414, 1.04866839736139433784435951726, 1.07532066234199239964883932544, 1.19015302654336680262086316331, 1.23606982104351875331330739030, 1.36891454208636501227539696840, 1.41086467593643744262265600826, 1.54286010113212644936170196913, 1.66299146394683237871553114404, 1.74357723956476554793062401648, 1.76824760168611945423313129837, 1.98933741999365350588236977846, 2.09748041020420391754801955811, 2.14326820126507998570725508745, 2.17784153476863363437713306591, 2.19296232636556929616015510186, 2.37971237986761298207558292979, 2.47977778246253514303772233892, 2.60200613790308956010116932702

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

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