Properties

Degree 24
Conductor $ 2^{60} \cdot 3^{12} \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  + 2·9-s + 48·19-s − 28·25-s − 8·27-s − 40·43-s − 6·49-s + 40·67-s − 72·73-s + 9·81-s − 56·97-s + 132·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 96·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s + 11.0·19-s − 5.59·25-s − 1.53·27-s − 6.09·43-s − 6/7·49-s + 4.88·67-s − 8.42·73-s + 81-s − 5.68·97-s + 12·121-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 + 2.15·169-s + 7.34·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{60} \cdot 3^{12} \cdot 7^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{672} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 2^{60} \cdot 3^{12} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $1.78380$
$L(\frac12)$  $\approx$  $1.78380$
$L(\frac{3}{2})$   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 \( ( 1 - T^{2} + 4 T^{3} - p T^{4} + p^{3} T^{6} )^{2} \)
7 \( ( 1 + T^{2} )^{6} \)
good5 \( ( 1 + 14 T^{2} + 107 T^{4} + 588 T^{6} + 107 p^{2} T^{8} + 14 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - p T^{2} )^{12} \)
13 \( ( 1 - 14 T^{2} + 331 T^{4} - 5132 T^{6} + 331 p^{2} T^{8} - 14 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( ( 1 - 38 T^{2} + 1151 T^{4} - 20724 T^{6} + 1151 p^{2} T^{8} - 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 - 12 T + 5 p T^{2} - 476 T^{3} + 5 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
23 \( ( 1 + 66 T^{2} + 2399 T^{4} + 64348 T^{6} + 2399 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 + 46 T^{2} + 2695 T^{4} + 76516 T^{6} + 2695 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - p T^{2} )^{12} \)
37 \( ( 1 - 94 T^{2} + 3191 T^{4} - 75972 T^{6} + 3191 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 86 T^{2} + 5647 T^{4} - 231764 T^{6} + 5647 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + 10 T + 3 p T^{2} + 844 T^{3} + 3 p^{2} T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
47 \( ( 1 + 58 T^{2} + 4591 T^{4} + 251884 T^{6} + 4591 p^{2} T^{8} + 58 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 + 62 T^{2} + 6551 T^{4} + 242436 T^{6} + 6551 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 226 T^{2} + 25867 T^{4} - 1888084 T^{6} + 25867 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 222 T^{2} + 25163 T^{4} - 1831724 T^{6} + 25163 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 10 T + 185 T^{2} - 1116 T^{3} + 185 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
71 \( ( 1 + 362 T^{2} + 58607 T^{4} + 5382060 T^{6} + 58607 p^{2} T^{8} + 362 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 + 6 T + p T^{2} )^{12} \)
79 \( ( 1 - 154 T^{2} + 18095 T^{4} - 1857324 T^{6} + 18095 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 162 T^{2} + 19931 T^{4} - 2214676 T^{6} + 19931 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 278 T^{2} + 40879 T^{4} - 4202324 T^{6} + 40879 p^{2} T^{8} - 278 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 + 14 T + 159 T^{2} + 1124 T^{3} + 159 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
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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

−3.34255373606098110823809476817, −3.34121715143151803048464949006, −3.29884625288680330731072295097, −3.10660813893704458908132572818, −3.01338555430615761330042736674, −2.94566501424293553125045185653, −2.93999047983456643756400939247, −2.93727101267638402690796849013, −2.75562286562231778517233645325, −2.47493295539165644922585310873, −2.36510530856770235044761467144, −2.25565002714985962923630914539, −1.92716004911322060342483326951, −1.89834086829333791644798490977, −1.74235581112556368978813290084, −1.74119985657418147964017472877, −1.72102239180814825761060095540, −1.47514841662586408117399565735, −1.36287198267682676292546522789, −1.24635816757981625207078040602, −1.10139391566579323654759940400, −0.875898040025905573793033105663, −0.844732997191226476839925604834, −0.38624735720477021309646381439, −0.12996123021008500749477993652, 0.12996123021008500749477993652, 0.38624735720477021309646381439, 0.844732997191226476839925604834, 0.875898040025905573793033105663, 1.10139391566579323654759940400, 1.24635816757981625207078040602, 1.36287198267682676292546522789, 1.47514841662586408117399565735, 1.72102239180814825761060095540, 1.74119985657418147964017472877, 1.74235581112556368978813290084, 1.89834086829333791644798490977, 1.92716004911322060342483326951, 2.25565002714985962923630914539, 2.36510530856770235044761467144, 2.47493295539165644922585310873, 2.75562286562231778517233645325, 2.93727101267638402690796849013, 2.93999047983456643756400939247, 2.94566501424293553125045185653, 3.01338555430615761330042736674, 3.10660813893704458908132572818, 3.29884625288680330731072295097, 3.34121715143151803048464949006, 3.34255373606098110823809476817

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

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