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Properties

Label 24-39e24-1.1-c3e12-0-0
Degree $24$
Conductor $1.533\times 10^{38}$
Sign $1$
Analytic cond. $2.72862\times 10^{23}$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $12$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·4-s + 494·16-s − 704·25-s − 1.62e3·43-s − 3.12e3·49-s − 2.58e3·61-s − 5.10e3·64-s − 6.38e3·79-s + 2.25e4·100-s + 5.48e3·103-s − 9.78e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.18e4·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 1.00e5·196-s + ⋯
L(s)  = 1  − 4·4-s + 7.71·16-s − 5.63·25-s − 5.74·43-s − 9.11·49-s − 5.43·61-s − 9.96·64-s − 9.08·79-s + 22.5·100-s + 5.24·103-s − 7.35·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 22.9·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 36.4·196-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{24} \cdot 13^{24}\)
Sign: $1$
Analytic conductor: \(2.72862\times 10^{23}\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(12\)
Selberg data: \((24,\ 3^{24} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( ( 1 + p^{4} T^{2} + 137 T^{4} + 467 p T^{6} + 137 p^{6} T^{8} + p^{16} T^{10} + p^{18} T^{12} )^{2} \)
5 \( ( 1 + 352 T^{2} + 67004 T^{4} + 9024514 T^{6} + 67004 p^{6} T^{8} + 352 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
7 \( ( 1 + 1563 T^{2} + 1133439 T^{4} + 490033226 T^{6} + 1133439 p^{6} T^{8} + 1563 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
11 \( ( 1 + 4894 T^{2} + 12501623 T^{4} + 20409809956 T^{6} + 12501623 p^{6} T^{8} + 4894 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
17 \( ( 1 + 17628 T^{2} + 9526812 p T^{4} + 976794215362 T^{6} + 9526812 p^{7} T^{8} + 17628 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
19 \( ( 1 + 31098 T^{2} + 454538055 T^{4} + 3954431343404 T^{6} + 454538055 p^{6} T^{8} + 31098 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
23 \( ( 1 + 12750 T^{2} + 342325839 T^{4} + 3933757293604 T^{6} + 342325839 p^{6} T^{8} + 12750 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
29 \( ( 1 + 31380 T^{2} + 605169732 T^{4} + 4204900932442 T^{6} + 605169732 p^{6} T^{8} + 31380 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 + 80727 T^{2} + 3957011115 T^{4} + 145324204911866 T^{6} + 3957011115 p^{6} T^{8} + 80727 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
37 \( ( 1 + 4632 p T^{2} + 14873563896 T^{4} + 882269012294426 T^{6} + 14873563896 p^{6} T^{8} + 4632 p^{13} T^{10} + p^{18} T^{12} )^{2} \)
41 \( ( 1 + 92032 T^{2} + 2129700188 T^{4} - 163503373356470 T^{6} + 2129700188 p^{6} T^{8} + 92032 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
43 \( ( 1 + 405 T + 218115 T^{2} + 52932278 T^{3} + 218115 p^{3} T^{4} + 405 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
47 \( ( 1 + 463894 T^{2} + 102299363711 T^{4} + 13443518873535124 T^{6} + 102299363711 p^{6} T^{8} + 463894 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
53 \( ( 1 + 123852 T^{2} + 20708854044 T^{4} + 6694409920272298 T^{6} + 20708854044 p^{6} T^{8} + 123852 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
59 \( ( 1 + 1063186 T^{2} + 496266110471 T^{4} + 131471882686878748 T^{6} + 496266110471 p^{6} T^{8} + 1063186 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
61 \( ( 1 + 647 T + 495110 T^{2} + 199819367 T^{3} + 495110 p^{3} T^{4} + 647 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
67 \( ( 1 + 1140171 T^{2} + 691356902343 T^{4} + 256566296538145946 T^{6} + 691356902343 p^{6} T^{8} + 1140171 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
71 \( ( 1 + 15338 p T^{2} + 458644803023 T^{4} + 140540566551693748 T^{6} + 458644803023 p^{6} T^{8} + 15338 p^{13} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 + 2222037 T^{2} + 2098500922278 T^{4} + 1077893830511252705 T^{6} + 2098500922278 p^{6} T^{8} + 2222037 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
79 \( ( 1 + 1595 T + 1551833 T^{2} + 1057580906 T^{3} + 1551833 p^{3} T^{4} + 1595 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
83 \( ( 1 + 1171294 T^{2} + 910614134951 T^{4} + 510677804747973604 T^{6} + 910614134951 p^{6} T^{8} + 1171294 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
89 \( ( 1 + 1411870 T^{2} + 1061285719727 T^{4} + 721505653517205508 T^{6} + 1061285719727 p^{6} T^{8} + 1411870 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
97 \( ( 1 + 2741019 T^{2} + 4046435660787 T^{4} + 4339635115970553266 T^{6} + 4046435660787 p^{6} T^{8} + 2741019 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
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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

−3.13534277347203690354847144154, −3.04807893999205490025972870354, −2.94910536182633937328749348977, −2.86108522880250945385885411756, −2.83544570507572525308767362836, −2.73122761614940507139521672273, −2.57400441918826570062017346824, −2.43960188685019649548119176478, −2.32736910913659985843228117489, −2.21622783226654669147075801708, −2.04118369732097907834709956259, −1.98956885589002649240866013484, −1.88783651777232653191646010217, −1.87375254654656882258203355812, −1.83162579089950489787386323964, −1.58889071502827004694130430808, −1.50338307624711691108476614246, −1.46238127519817697733532511664, −1.30430836212901873587652028108, −1.21835728511680407279552464207, −1.20310557666533047211696675813, −1.11306478795608159847115203144, −1.03484266777324195215219291139, −0.962251394750458867003509367429, −0.921245835721242746450834554510, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.921245835721242746450834554510, 0.962251394750458867003509367429, 1.03484266777324195215219291139, 1.11306478795608159847115203144, 1.20310557666533047211696675813, 1.21835728511680407279552464207, 1.30430836212901873587652028108, 1.46238127519817697733532511664, 1.50338307624711691108476614246, 1.58889071502827004694130430808, 1.83162579089950489787386323964, 1.87375254654656882258203355812, 1.88783651777232653191646010217, 1.98956885589002649240866013484, 2.04118369732097907834709956259, 2.21622783226654669147075801708, 2.32736910913659985843228117489, 2.43960188685019649548119176478, 2.57400441918826570062017346824, 2.73122761614940507139521672273, 2.83544570507572525308767362836, 2.86108522880250945385885411756, 2.94910536182633937328749348977, 3.04807893999205490025972870354, 3.13534277347203690354847144154

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

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