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  + 24·13-s − 12·25-s − 6·49-s − 72·61-s − 72·113-s − 60·121-s − 48·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 156·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 6.65·13-s − 2.39·25-s − 6/7·49-s − 9.21·61-s − 6.77·113-s − 5.45·121-s − 4.29·125-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 + 12·169-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 + ⋯

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  =  $(24,\ 2^{72} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )$
$L(1)$  $\approx$  $0.01293923525$
$L(\frac12)$  $\approx$  $0.01293923525$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 24. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 23.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + 6 T^{2} - 33 T^{4} - 556 T^{6} - 33 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \)
good5 \( ( 1 + 3 T^{2} + 12 T^{3} + 3 p T^{4} + p^{3} T^{6} )^{4} \)
11 \( ( 1 + 30 T^{2} + 327 T^{4} + 2548 T^{6} + 327 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 2 T + p T^{2} )^{12} \)
17 \( ( 1 - 30 T^{2} + 831 T^{4} - 12628 T^{6} + 831 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 - 66 T^{2} + 2343 T^{4} - 54844 T^{6} + 2343 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
23 \( ( 1 - 78 T^{2} + 3567 T^{4} - 98836 T^{6} + 3567 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 42 T^{2} + 1575 T^{4} - 60076 T^{6} + 1575 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
31 \( ( 1 + 54 T^{2} + 3471 T^{4} + 101684 T^{6} + 3471 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 126 T^{2} + 7863 T^{4} - 337412 T^{6} + 7863 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 126 T^{2} + 6639 T^{4} - 258580 T^{6} + 6639 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 + 78 T^{2} + 4503 T^{4} + 248420 T^{6} + 4503 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( ( 1 + 90 T^{2} + 7791 T^{4} + 391852 T^{6} + 7791 p^{2} T^{8} + 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 186 T^{2} + 19575 T^{4} - 1258444 T^{6} + 19575 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 258 T^{2} + 31863 T^{4} - 2365180 T^{6} + 31863 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 18 T + 195 T^{2} + 1484 T^{3} + 195 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
67 \( ( 1 + 78 T^{2} + 3399 T^{4} + 64676 T^{6} + 3399 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 222 T^{2} + 23727 T^{4} - 1839796 T^{6} + 23727 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 150 T^{2} + 16575 T^{4} - 1350452 T^{6} + 16575 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 234 T^{2} + 22767 T^{4} - 1649932 T^{6} + 22767 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 258 T^{2} + 42087 T^{4} - 4123708 T^{6} + 42087 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 222 T^{2} + 32463 T^{4} - 3429460 T^{6} + 32463 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 - 390 T^{2} + 77391 T^{4} - 9349652 T^{6} + 77391 p^{2} T^{8} - 390 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−2.57867399557314540475117732927, −2.57281981010476294184996595781, −2.41236765596003841162529901204, −2.27953161796830063214484788464, −2.09765569965617703909680316887, −2.05502858326835760358167445805, −1.98667923812429529081835840817, −1.90711247947262028754099243888, −1.78420114939453721766487060887, −1.59936073333356926646443002196, −1.56042300046404868427581920327, −1.50502126150712678040164428497, −1.42889940851340566984453041599, −1.37576482807414501065882195864, −1.27083299168847145997356186623, −1.24307902619873227175361245883, −1.19369671947814296886386512867, −1.09008592049863520992888098206, −1.05194115337305696251236064439, −0.982930203167179147124353260843, −0.66263600220072627153265327579, −0.29376694343137726575926845960, −0.26332106281570304303525063134, −0.24594965859372605432813499733, −0.008426563678264835924467125543, 0.008426563678264835924467125543, 0.24594965859372605432813499733, 0.26332106281570304303525063134, 0.29376694343137726575926845960, 0.66263600220072627153265327579, 0.982930203167179147124353260843, 1.05194115337305696251236064439, 1.09008592049863520992888098206, 1.19369671947814296886386512867, 1.24307902619873227175361245883, 1.27083299168847145997356186623, 1.37576482807414501065882195864, 1.42889940851340566984453041599, 1.50502126150712678040164428497, 1.56042300046404868427581920327, 1.59936073333356926646443002196, 1.78420114939453721766487060887, 1.90711247947262028754099243888, 1.98667923812429529081835840817, 2.05502858326835760358167445805, 2.09765569965617703909680316887, 2.27953161796830063214484788464, 2.41236765596003841162529901204, 2.57281981010476294184996595781, 2.57867399557314540475117732927

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

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