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Properties

Label 24-240e12-1.1-c2e12-0-0
Degree $24$
Conductor $3.652\times 10^{28}$
Sign $1$
Analytic cond. $6.11724\times 10^{9}$
Root an. cond. $2.55724$
Motivic weight $2$
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  + 4·9-s + 18·25-s + 48·31-s + 168·49-s + 144·61-s − 432·79-s − 39·81-s − 624·109-s + 780·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 492·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 4/9·9-s + 0.719·25-s + 1.54·31-s + 24/7·49-s + 2.36·61-s − 5.46·79-s − 0.481·81-s − 5.72·109-s + 6.44·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.91·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 3^{12} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(6.11724\times 10^{9}\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 3^{12} \cdot 5^{12} ,\ ( \ : [1]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8611083572\)
\(L(\frac12)\) \(\approx\) \(0.8611083572\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 4 T^{2} + 55 T^{4} + 56 p^{2} T^{6} + 55 p^{4} T^{8} - 4 p^{8} T^{10} + p^{12} T^{12} \)
5 \( 1 - 18 T^{2} - 369 T^{4} + 36 p^{4} T^{6} - 369 p^{4} T^{8} - 18 p^{8} T^{10} + p^{12} T^{12} \)
good7 \( ( 1 - 12 p T^{2} + 2535 T^{4} - 76264 T^{6} + 2535 p^{4} T^{8} - 12 p^{9} T^{10} + p^{12} T^{12} )^{2} \)
11 \( ( 1 - 390 T^{2} + 91935 T^{4} - 13288020 T^{6} + 91935 p^{4} T^{8} - 390 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
13 \( ( 1 - 246 T^{2} + 67071 T^{4} - 11718004 T^{6} + 67071 p^{4} T^{8} - 246 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
17 \( ( 1 + 1470 T^{2} + 961599 T^{4} + 358361732 T^{6} + 961599 p^{4} T^{8} + 1470 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
19 \( ( 1 + 303 T^{2} - 5648 T^{3} + 303 p^{2} T^{4} + p^{6} T^{6} )^{4} \)
23 \( ( 1 + 2124 T^{2} + 2265255 T^{4} + 1490798808 T^{6} + 2265255 p^{4} T^{8} + 2124 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
29 \( ( 1 - 630 T^{2} + 49599 T^{4} + 425800332 T^{6} + 49599 p^{4} T^{8} - 630 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
31 \( ( 1 - 12 T + 1191 T^{2} + 8040 T^{3} + 1191 p^{2} T^{4} - 12 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
37 \( ( 1 - 5622 T^{2} + 15747615 T^{4} - 26937456628 T^{6} + 15747615 p^{4} T^{8} - 5622 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
41 \( ( 1 - 5730 T^{2} + 17922783 T^{4} - 36601296060 T^{6} + 17922783 p^{4} T^{8} - 5730 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
43 \( ( 1 - 7044 T^{2} + 25527255 T^{4} - 57654006536 T^{6} + 25527255 p^{4} T^{8} - 7044 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
47 \( ( 1 + 7500 T^{2} + 30379143 T^{4} + 80184503000 T^{6} + 30379143 p^{4} T^{8} + 7500 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
53 \( ( 1 + 5454 T^{2} + 24601167 T^{4} + 74943076644 T^{6} + 24601167 p^{4} T^{8} + 5454 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
59 \( ( 1 - 9030 T^{2} + 39008991 T^{4} - 135788124116 T^{6} + 39008991 p^{4} T^{8} - 9030 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
61 \( ( 1 - 36 T + 10815 T^{2} - 265928 T^{3} + 10815 p^{2} T^{4} - 36 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
67 \( ( 1 - 21828 T^{2} + 215026743 T^{4} - 1231707761032 T^{6} + 215026743 p^{4} T^{8} - 21828 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
71 \( ( 1 - 11046 T^{2} + 99426735 T^{4} - 572370011988 T^{6} + 99426735 p^{4} T^{8} - 11046 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
73 \( ( 1 - 5574 T^{2} + 26630895 T^{4} - 31585511956 T^{6} + 26630895 p^{4} T^{8} - 5574 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
79 \( ( 1 + 108 T + 15207 T^{2} + 1113656 T^{3} + 15207 p^{2} T^{4} + 108 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
83 \( ( 1 + 26748 T^{2} + 362457207 T^{4} + 3108598049016 T^{6} + 362457207 p^{4} T^{8} + 26748 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
89 \( ( 1 - 19878 T^{2} + 233842479 T^{4} - 1925486462036 T^{6} + 233842479 p^{4} T^{8} - 19878 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
97 \( ( 1 - 43110 T^{2} + 827297295 T^{4} - 9601013122132 T^{6} + 827297295 p^{4} T^{8} - 43110 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
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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

−4.07220006359574597880419048412, −4.01465032341126409334536682949, −3.68772097564330456928486313712, −3.36515458633466123328605587338, −3.36138716069819525911318356681, −3.33616271212771172498748246617, −3.30990185709271309187932579374, −3.08534475088008722864632223255, −2.95519425305292711617481044833, −2.78208952184208912452297363544, −2.69305804150334810322208699481, −2.52660127541498326016691504960, −2.41405479214756510047495721177, −2.22459944930654995620920580826, −2.16360506762907009292377625240, −2.08959042348337828766990075509, −1.78222536992214321262451007784, −1.70886955074925118310841616316, −1.30395879449420470822513426213, −1.13153909278530656020438027361, −1.08080023576183287156607738105, −1.04354724448697656679772341612, −0.832475359297004753541465432870, −0.34285760598174426797576369275, −0.089767550623255913970475763361, 0.089767550623255913970475763361, 0.34285760598174426797576369275, 0.832475359297004753541465432870, 1.04354724448697656679772341612, 1.08080023576183287156607738105, 1.13153909278530656020438027361, 1.30395879449420470822513426213, 1.70886955074925118310841616316, 1.78222536992214321262451007784, 2.08959042348337828766990075509, 2.16360506762907009292377625240, 2.22459944930654995620920580826, 2.41405479214756510047495721177, 2.52660127541498326016691504960, 2.69305804150334810322208699481, 2.78208952184208912452297363544, 2.95519425305292711617481044833, 3.08534475088008722864632223255, 3.30990185709271309187932579374, 3.33616271212771172498748246617, 3.36138716069819525911318356681, 3.36515458633466123328605587338, 3.68772097564330456928486313712, 4.01465032341126409334536682949, 4.07220006359574597880419048412

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

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