Properties

Label 24-819e12-1.1-c1e12-0-6
Degree $24$
Conductor $9.108\times 10^{34}$
Sign $1$
Analytic cond. $6.11972\times 10^{9}$
Root an. cond. $2.55729$
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  + 4·2-s + 8·4-s − 8·7-s + 12·8-s + 8·11-s − 32·14-s + 20·16-s + 32·22-s − 64·28-s + 4·29-s + 40·32-s + 12·37-s + 64·44-s + 32·49-s + 12·53-s − 96·56-s + 16·58-s + 72·64-s + 60·67-s + 48·74-s − 64·77-s − 4·79-s + 96·88-s + 128·98-s + 48·106-s − 88·107-s − 56·109-s + ⋯
L(s)  = 1  + 2.82·2-s + 4·4-s − 3.02·7-s + 4.24·8-s + 2.41·11-s − 8.55·14-s + 5·16-s + 6.82·22-s − 12.0·28-s + 0.742·29-s + 7.07·32-s + 1.97·37-s + 9.64·44-s + 32/7·49-s + 1.64·53-s − 12.8·56-s + 2.10·58-s + 9·64-s + 7.33·67-s + 5.57·74-s − 7.29·77-s − 0.450·79-s + 10.2·88-s + 12.9·98-s + 4.66·106-s − 8.50·107-s − 5.36·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{24} \cdot 7^{12} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(6.11972\times 10^{9}\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{819} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{24} \cdot 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(23.95591436\)
\(L(\frac12)\) \(\approx\) \(23.95591436\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + 8 T + 32 T^{2} + 72 T^{3} - 13 T^{4} - 704 T^{5} - 2624 T^{6} - 704 p T^{7} - 13 p^{2} T^{8} + 72 p^{3} T^{9} + 32 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 - 20 T^{2} - 165 T^{4} + 40 p^{2} T^{6} - 165 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} \)
good2 \( ( 1 - p T + p T^{2} - p T^{3} + p T^{6} - p^{4} T^{9} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} )^{2} \)
5 \( 1 + T^{4} + 94 T^{8} - 18431 T^{12} + 94 p^{4} T^{16} + p^{8} T^{20} + p^{12} T^{24} \)
11 \( ( 1 - 4 T + 8 T^{2} - 46 T^{3} + 3 p^{2} T^{4} - 82 p T^{5} + 1762 T^{6} - 82 p^{2} T^{7} + 3 p^{4} T^{8} - 46 p^{3} T^{9} + 8 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
17 \( ( 1 + 20 T^{2} + 445 T^{4} + 4010 T^{6} + 445 p^{2} T^{8} + 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
19 \( 1 - 355 T^{4} - 95870 T^{8} + 114782513 T^{12} - 95870 p^{4} T^{16} - 355 p^{8} T^{20} + p^{12} T^{24} \)
23 \( ( 1 - 75 T^{2} + 3030 T^{4} - 83635 T^{6} + 3030 p^{2} T^{8} - 75 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 - T + 54 T^{2} + 27 T^{3} + 54 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{4} \)
31 \( 1 - 1423 T^{4} + 233254 T^{8} + 417094841 T^{12} + 233254 p^{4} T^{16} - 1423 p^{8} T^{20} + p^{12} T^{24} \)
37 \( ( 1 - 6 T + 18 T^{2} - 224 T^{3} + 2987 T^{4} - 9502 T^{5} + 28334 T^{6} - 9502 p T^{7} + 2987 p^{2} T^{8} - 224 p^{3} T^{9} + 18 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
41 \( 1 + 6966 T^{4} + 24349871 T^{8} + 51186333364 T^{12} + 24349871 p^{4} T^{16} + 6966 p^{8} T^{20} + p^{12} T^{24} \)
43 \( ( 1 - 63 T^{2} + 4310 T^{4} - 213179 T^{6} + 4310 p^{2} T^{8} - 63 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( 1 - 4499 T^{4} + 232910 p T^{8} - 19979553119 T^{12} + 232910 p^{5} T^{16} - 4499 p^{8} T^{20} + p^{12} T^{24} \)
53 \( ( 1 - 3 T + 104 T^{2} - 431 T^{3} + 104 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
59 \( 1 + 8826 T^{4} + 20574431 T^{8} + 13508299324 T^{12} + 20574431 p^{4} T^{16} + 8826 p^{8} T^{20} + p^{12} T^{24} \)
61 \( ( 1 - 170 T^{2} + 20487 T^{4} - 1428236 T^{6} + 20487 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
67 \( ( 1 - 30 T + 450 T^{2} - 5098 T^{3} + 47495 T^{4} - 5660 p T^{5} + 668 p^{2} T^{6} - 5660 p^{2} T^{7} + 47495 p^{2} T^{8} - 5098 p^{3} T^{9} + 450 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
71 \( ( 1 - 134 T^{3} + 3423 T^{4} + 21842 T^{5} + 8978 T^{6} + 21842 p T^{7} + 3423 p^{2} T^{8} - 134 p^{3} T^{9} + p^{6} T^{12} )^{2} \)
73 \( 1 - 8035 T^{4} - 27974390 T^{8} + 458366283353 T^{12} - 27974390 p^{4} T^{16} - 8035 p^{8} T^{20} + p^{12} T^{24} \)
79 \( ( 1 + T - 24 T^{2} - 1007 T^{3} - 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( 1 - 2655 T^{4} + 72855350 T^{8} - 404915929367 T^{12} + 72855350 p^{4} T^{16} - 2655 p^{8} T^{20} + p^{12} T^{24} \)
89 \( 1 - 3395 T^{4} + 117237082 T^{8} - 235034612951 T^{12} + 117237082 p^{4} T^{16} - 3395 p^{8} T^{20} + p^{12} T^{24} \)
97 \( 1 + 19557 T^{4} + 258666194 T^{8} + 3056968310041 T^{12} + 258666194 p^{4} T^{16} + 19557 p^{8} T^{20} + p^{12} T^{24} \)
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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

−3.46210904177348170202626013314, −3.43864856068506250095543820609, −3.20743431596971071316108314821, −2.89640849305034891851334122362, −2.87719295486086940904754701810, −2.81013640333195791431162422317, −2.79123363959967891980021619094, −2.68574889199557629581800307476, −2.66920047335868758302111588595, −2.54496383292511640867257176254, −2.35449943855742302615763014396, −2.25125755686320656244593216155, −2.15677309158733050738340063243, −2.10336824675040336954284860000, −2.06396097415801544588797427015, −1.59547213440903302772086212493, −1.43492746388035678306068519713, −1.39915914005391202519108058685, −1.32430378115727671154825311617, −1.22072391266504647233698009534, −1.04194100177051884101677690715, −0.836127034051103610379332430558, −0.801199029837687554194185906222, −0.41173112437466636605385971954, −0.20900037132789734845765172446, 0.20900037132789734845765172446, 0.41173112437466636605385971954, 0.801199029837687554194185906222, 0.836127034051103610379332430558, 1.04194100177051884101677690715, 1.22072391266504647233698009534, 1.32430378115727671154825311617, 1.39915914005391202519108058685, 1.43492746388035678306068519713, 1.59547213440903302772086212493, 2.06396097415801544588797427015, 2.10336824675040336954284860000, 2.15677309158733050738340063243, 2.25125755686320656244593216155, 2.35449943855742302615763014396, 2.54496383292511640867257176254, 2.66920047335868758302111588595, 2.68574889199557629581800307476, 2.79123363959967891980021619094, 2.81013640333195791431162422317, 2.87719295486086940904754701810, 2.89640849305034891851334122362, 3.20743431596971071316108314821, 3.43864856068506250095543820609, 3.46210904177348170202626013314

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

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