Properties

Degree 24
Conductor $ 2^{36} \cdot 3^{12} \cdot 5^{24} $
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  − 4-s − 6·9-s + 16-s − 32·31-s + 6·36-s − 8·41-s − 36·49-s + 7·64-s + 32·71-s − 32·79-s + 21·81-s + 40·89-s + 68·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + 151-s + 157-s + 163-s + 8·164-s + 167-s + 60·169-s + 173-s + ⋯
L(s)  = 1  − 1/2·4-s − 2·9-s + 1/4·16-s − 5.74·31-s + 36-s − 1.24·41-s − 5.14·49-s + 7/8·64-s + 3.79·71-s − 3.60·79-s + 7/3·81-s + 4.23·89-s + 6.18·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.624·164-s + 0.0773·167-s + 4.61·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{36} \cdot 3^{12} \cdot 5^{24}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((24,\ 2^{36} \cdot 3^{12} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(1.65779\)
\(L(\frac12)\)  \(\approx\)  \(1.65779\)
\(L(\frac{3}{2})\)   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 24. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( 1 + T^{2} - p^{3} T^{6} + p^{4} T^{10} + p^{6} T^{12} \)
3 \( ( 1 + T^{2} )^{6} \)
5 \( 1 \)
good7 \( ( 1 + 18 T^{2} + 191 T^{4} + 1532 T^{6} + 191 p^{2} T^{8} + 18 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 30 T^{2} + 743 T^{4} - 10436 T^{6} + 743 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 66 T^{2} + 2255 T^{4} + 47324 T^{6} + 2255 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
23 \( ( 1 + 2 p T^{2} + 1775 T^{4} + 40932 T^{6} + 1775 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
37 \( ( 1 - 158 T^{2} + 11191 T^{4} - 496772 T^{6} + 11191 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
43 \( ( 1 - 130 T^{2} + 9815 T^{4} - 518268 T^{6} + 9815 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 222 T^{2} + 22367 T^{4} + 1328324 T^{6} + 22367 p^{2} T^{8} + 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 238 T^{2} + 26391 T^{4} - 1758052 T^{6} + 26391 p^{2} T^{8} - 238 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 274 T^{2} + 37127 T^{4} - 3112476 T^{6} + 37127 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
73 \( ( 1 + 54 T^{2} + 2367 T^{4} + 531700 T^{6} + 2367 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 233 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 - 306 T^{2} + 50855 T^{4} - 5168732 T^{6} + 50855 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 10 T + 103 T^{2} - 396 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
97 \( ( 1 + 246 T^{2} + 39183 T^{4} + 4535476 T^{6} + 39183 p^{2} T^{8} + 246 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−3.50054771203984655960273502930, −3.36937023508344912732079935475, −3.29917975632862668681258457054, −3.29280473848373169082831061709, −3.19194387930157377635838153012, −3.09348413190969811732507335812, −2.82148910602230004932074902259, −2.80989517177547999257009869207, −2.72535109197844120118420219990, −2.68012783790873851594149227982, −2.39050774276089377667442248232, −2.18850875419718800536783374044, −2.05918041610614081917854046339, −1.99516143229671399088654309103, −1.90867234337373783480437879351, −1.88015672129122475097908214960, −1.83098214645737332572255802108, −1.63079743699465119218794675979, −1.51681058281702573856635576134, −1.31237399614592677664554219641, −0.888008447718660803343565561406, −0.881974301390377661719713125968, −0.47502047449645963913180043947, −0.36412827931720773684822848800, −0.28414314521316499865016354011, 0.28414314521316499865016354011, 0.36412827931720773684822848800, 0.47502047449645963913180043947, 0.881974301390377661719713125968, 0.888008447718660803343565561406, 1.31237399614592677664554219641, 1.51681058281702573856635576134, 1.63079743699465119218794675979, 1.83098214645737332572255802108, 1.88015672129122475097908214960, 1.90867234337373783480437879351, 1.99516143229671399088654309103, 2.05918041610614081917854046339, 2.18850875419718800536783374044, 2.39050774276089377667442248232, 2.68012783790873851594149227982, 2.72535109197844120118420219990, 2.80989517177547999257009869207, 2.82148910602230004932074902259, 3.09348413190969811732507335812, 3.19194387930157377635838153012, 3.29280473848373169082831061709, 3.29917975632862668681258457054, 3.36937023508344912732079935475, 3.50054771203984655960273502930

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

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