Properties

Label 24-483e12-1.1-c1e12-0-2
Degree $24$
Conductor $1.612\times 10^{32}$
Sign $1$
Analytic cond. $1.08315\times 10^{7}$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 4·8-s − 6·9-s + 24·16-s − 16·23-s − 18·25-s + 16·29-s + 40·32-s + 48·36-s − 8·49-s − 22·64-s + 76·71-s + 24·72-s + 21·81-s + 128·92-s + 144·100-s − 128·116-s + 104·121-s + 127-s − 160·128-s + 131-s + 137-s + 139-s − 144·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 4·4-s − 1.41·8-s − 2·9-s + 6·16-s − 3.33·23-s − 3.59·25-s + 2.97·29-s + 7.07·32-s + 8·36-s − 8/7·49-s − 2.75·64-s + 9.01·71-s + 2.82·72-s + 7/3·81-s + 13.3·92-s + 72/5·100-s − 11.8·116-s + 9.45·121-s + 0.0887·127-s − 14.1·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 12·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{12} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 7^{12} \cdot 23^{12}\)
Sign: $1$
Analytic conductor: \(1.08315\times 10^{7}\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{483} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 7^{12} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2496027822\)
\(L(\frac12)\) \(\approx\) \(0.2496027822\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T^{2} )^{6} \)
7 \( 1 + 8 T^{2} - 33 T^{4} - 16 p^{2} T^{6} - 33 p^{2} T^{8} + 8 p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 8 T - 19 T^{2} - 344 T^{3} - 19 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good2 \( ( 1 + p T^{2} + T^{3} + p^{2} T^{4} + p^{3} T^{6} )^{4} \)
5 \( ( 1 + 9 T^{2} + 26 T^{4} + 46 T^{6} + 26 p^{2} T^{8} + 9 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - 52 T^{2} + 1252 T^{4} - 17566 T^{6} + 1252 p^{2} T^{8} - 52 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
13 \( ( 1 - 31 T^{2} + 736 T^{4} - 10936 T^{6} + 736 p^{2} T^{8} - 31 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 50 T^{2} + 880 T^{4} + 10832 T^{6} + 880 p^{2} T^{8} + 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( ( 1 + 30 T^{2} + 1272 T^{4} + 21228 T^{6} + 1272 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - 4 T + 80 T^{2} - 206 T^{3} + 80 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
31 \( ( 1 - 40 T^{2} + 2200 T^{4} - 53566 T^{6} + 2200 p^{2} T^{8} - 40 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( ( 1 - 56 T^{2} + 4908 T^{4} - 153866 T^{6} + 4908 p^{2} T^{8} - 56 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 - 88 T^{2} + 4904 T^{4} - 193038 T^{6} + 4904 p^{2} T^{8} - 88 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( ( 1 - 17 T^{2} + 714 T^{4} + 82702 T^{6} + 714 p^{2} T^{8} - 17 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( ( 1 - 106 T^{2} + 9455 T^{4} - 474444 T^{6} + 9455 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 101 T^{2} + 194 p T^{4} - 549734 T^{6} + 194 p^{3} T^{8} - 101 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 - 235 T^{2} + 28136 T^{4} - 2061612 T^{6} + 28136 p^{2} T^{8} - 235 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
61 \( ( 1 + 247 T^{2} + 28272 T^{4} + 2062228 T^{6} + 28272 p^{2} T^{8} + 247 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 261 T^{2} + 34346 T^{4} - 2821306 T^{6} + 34346 p^{2} T^{8} - 261 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 19 T + 318 T^{2} - 2832 T^{3} + 318 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
73 \( ( 1 - 324 T^{2} + 49896 T^{4} - 4606954 T^{6} + 49896 p^{2} T^{8} - 324 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 150 T^{2} + 19856 T^{4} - 1516852 T^{6} + 19856 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
83 \( ( 1 + 444 T^{2} + 86276 T^{4} + 9344374 T^{6} + 86276 p^{2} T^{8} + 444 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 257 T^{2} + 25562 T^{4} + 1888566 T^{6} + 25562 p^{2} T^{8} + 257 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
97 \( ( 1 + 208 T^{2} + 18228 T^{4} + 1196050 T^{6} + 18228 p^{2} T^{8} + 208 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.70845310422088645837844062269, −3.69839733224046163664050344422, −3.60815073457334114195230687116, −3.31076798953150668395354237144, −3.27722170801616874437916166542, −3.16135255178487280139185860636, −3.15474729685199284979089818805, −2.99338532709885610225481468324, −2.83715095265614644966998497207, −2.59320247479866846798542933645, −2.50684068057011214115807020072, −2.44939828587195326512387526671, −2.32578866399791171734492496739, −2.14048648471998984580123345456, −2.09500060673471962157198010144, −2.02612145654218513515895664373, −1.93490868601404943841463345237, −1.58775319213468803849386458630, −1.53335535957880372390536042589, −1.20636896531278492666254011965, −0.75318145690819381383556658364, −0.72835674730361729254504643229, −0.51155809807403060200635116245, −0.42466584081625905436003491518, −0.22025012169095997767510772350, 0.22025012169095997767510772350, 0.42466584081625905436003491518, 0.51155809807403060200635116245, 0.72835674730361729254504643229, 0.75318145690819381383556658364, 1.20636896531278492666254011965, 1.53335535957880372390536042589, 1.58775319213468803849386458630, 1.93490868601404943841463345237, 2.02612145654218513515895664373, 2.09500060673471962157198010144, 2.14048648471998984580123345456, 2.32578866399791171734492496739, 2.44939828587195326512387526671, 2.50684068057011214115807020072, 2.59320247479866846798542933645, 2.83715095265614644966998497207, 2.99338532709885610225481468324, 3.15474729685199284979089818805, 3.16135255178487280139185860636, 3.27722170801616874437916166542, 3.31076798953150668395354237144, 3.60815073457334114195230687116, 3.69839733224046163664050344422, 3.70845310422088645837844062269

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

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