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Properties

Label 24-975e12-1.1-c1e12-0-7
Degree $24$
Conductor $7.380\times 10^{35}$
Sign $1$
Analytic cond. $4.95884\times 10^{10}$
Root an. cond. $2.79023$
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  + 3·9-s + 24·19-s + 12·29-s + 12·31-s − 27·49-s − 12·59-s + 6·61-s + 24·64-s + 24·71-s + 12·79-s + 3·81-s + 60·89-s + 48·101-s − 108·109-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 21·169-s + 72·171-s + 173-s + ⋯
L(s)  = 1  + 9-s + 5.50·19-s + 2.22·29-s + 2.15·31-s − 3.85·49-s − 1.56·59-s + 0.768·61-s + 3·64-s + 2.84·71-s + 1.35·79-s + 1/3·81-s + 6.35·89-s + 4.77·101-s − 10.3·109-s + 1.63·121-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 − 1.61·169-s + 5.50·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 5^{24} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(4.95884\times 10^{10}\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 5^{24} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(34.26533170\)
\(L(\frac12)\) \(\approx\) \(34.26533170\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T^{2} + T^{4} )^{3} \)
5 \( 1 \)
13 \( 1 + 21 T^{2} + 378 T^{4} + 4601 T^{6} + 378 p^{2} T^{8} + 21 p^{4} T^{10} + p^{6} T^{12} \)
good2 \( ( 1 - 3 p^{2} T^{6} + p^{6} T^{12} )^{2} \)
7 \( 1 + 27 T^{2} + 387 T^{4} + 3596 T^{6} + 24093 T^{8} + 123993 T^{10} + 700494 T^{12} + 123993 p^{2} T^{14} + 24093 p^{4} T^{16} + 3596 p^{6} T^{18} + 387 p^{8} T^{20} + 27 p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 - 9 T^{2} + 32 T^{3} - 18 T^{4} - 144 T^{5} + 2027 T^{6} - 144 p T^{7} - 18 p^{2} T^{8} + 32 p^{3} T^{9} - 9 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 18 T^{2} - 63 T^{4} + 1014 T^{6} - 15642 T^{8} - 1371258 T^{10} - 13212331 T^{12} - 1371258 p^{2} T^{14} - 15642 p^{4} T^{16} + 1014 p^{6} T^{18} - 63 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 - 12 T + 51 T^{2} - 196 T^{3} + 1362 T^{4} - 4704 T^{5} + 9003 T^{6} - 4704 p T^{7} + 1362 p^{2} T^{8} - 196 p^{3} T^{9} + 51 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 42 T^{2} + 357 T^{4} - 11714 T^{6} - 251622 T^{8} - 954702 T^{10} + 8641149 T^{12} - 954702 p^{2} T^{14} - 251622 p^{4} T^{16} - 11714 p^{6} T^{18} + 357 p^{8} T^{20} + 42 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 6 T - 51 T^{2} + 146 T^{3} + 3042 T^{4} - 132 p T^{5} - 89443 T^{6} - 132 p^{2} T^{7} + 3042 p^{2} T^{8} + 146 p^{3} T^{9} - 51 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 3 T + 177 T^{3} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
37 \( 1 + 114 T^{2} + 7245 T^{4} + 226246 T^{6} + 726906 T^{8} - 346769094 T^{10} - 18414231051 T^{12} - 346769094 p^{2} T^{14} + 726906 p^{4} T^{16} + 226246 p^{6} T^{18} + 7245 p^{8} T^{20} + 114 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 - 99 T^{2} + 52 T^{3} + 5742 T^{4} - 2574 T^{5} - 263323 T^{6} - 2574 p T^{7} + 5742 p^{2} T^{8} + 52 p^{3} T^{9} - 99 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
43 \( 1 + 207 T^{2} + 23463 T^{4} + 1891696 T^{6} + 120427461 T^{8} + 6428829969 T^{10} + 296143620054 T^{12} + 6428829969 p^{2} T^{14} + 120427461 p^{4} T^{16} + 1891696 p^{6} T^{18} + 23463 p^{8} T^{20} + 207 p^{10} T^{22} + p^{12} T^{24} \)
47 \( ( 1 - 126 T^{2} + 8787 T^{4} - 434520 T^{6} + 8787 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 270 T^{2} + 32535 T^{4} - 2229348 T^{6} + 32535 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
59 \( ( 1 + 6 T - 129 T^{2} - 398 T^{3} + 13326 T^{4} + 18576 T^{5} - 839197 T^{6} + 18576 p T^{7} + 13326 p^{2} T^{8} - 398 p^{3} T^{9} - 129 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 3 T - 153 T^{2} + 112 T^{3} + 15465 T^{4} - 309 T^{5} - 1091010 T^{6} - 309 p T^{7} + 15465 p^{2} T^{8} + 112 p^{3} T^{9} - 153 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 159 T^{2} + 10191 T^{4} + 253544 T^{6} - 10619307 T^{8} - 2147172471 T^{10} - 199627454874 T^{12} - 2147172471 p^{2} T^{14} - 10619307 p^{4} T^{16} + 253544 p^{6} T^{18} + 10191 p^{8} T^{20} + 159 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 12 T - 21 T^{2} + 64 T^{3} + 3246 T^{4} + 52038 T^{5} - 898573 T^{6} + 52038 p T^{7} + 3246 p^{2} T^{8} + 64 p^{3} T^{9} - 21 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( ( 1 - 183 T^{2} + 17862 T^{4} - 1427551 T^{6} + 17862 p^{2} T^{8} - 183 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 3 T + 144 T^{2} - 111 T^{3} + 144 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 - 294 T^{2} + 45255 T^{4} - 4576948 T^{6} + 45255 p^{2} T^{8} - 294 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 30 T + 345 T^{2} - 4206 T^{3} + 71970 T^{4} - 727140 T^{5} + 5580097 T^{6} - 727140 p T^{7} + 71970 p^{2} T^{8} - 4206 p^{3} T^{9} + 345 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 + 339 T^{2} + 50319 T^{4} + 6636256 T^{6} + 978360357 T^{8} + 1143633741 p T^{10} + 10470442256742 T^{12} + 1143633741 p^{3} T^{14} + 978360357 p^{4} T^{16} + 6636256 p^{6} T^{18} + 50319 p^{8} T^{20} + 339 p^{10} T^{22} + p^{12} T^{24} \)
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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

−3.24674561194127173003622621102, −3.24490275868073548859918395360, −3.18584403962802770143066348153, −3.06410809706778392471009430256, −2.73599594750521509939223006524, −2.71732740632292555139670898650, −2.54882305049903936423742164354, −2.47028498583580453783136411206, −2.42306448433730843991477163302, −2.36641187441258732899647569051, −2.22888272492982428661049423375, −2.15609213310718831549318228313, −1.95864931861293243994106904342, −1.77802606313867218199500113497, −1.65974582626602317774906310468, −1.63854033343728516410373469507, −1.46850725576020344863727632897, −1.19515049846555443343075281532, −1.14535903776205254175578664103, −1.11585270968088732754125548874, −0.910380409569910204557259302138, −0.78701785559276411379273903530, −0.74539359309409158811298267110, −0.50784747088814196214706363586, −0.34497033916092636480424249481, 0.34497033916092636480424249481, 0.50784747088814196214706363586, 0.74539359309409158811298267110, 0.78701785559276411379273903530, 0.910380409569910204557259302138, 1.11585270968088732754125548874, 1.14535903776205254175578664103, 1.19515049846555443343075281532, 1.46850725576020344863727632897, 1.63854033343728516410373469507, 1.65974582626602317774906310468, 1.77802606313867218199500113497, 1.95864931861293243994106904342, 2.15609213310718831549318228313, 2.22888272492982428661049423375, 2.36641187441258732899647569051, 2.42306448433730843991477163302, 2.47028498583580453783136411206, 2.54882305049903936423742164354, 2.71732740632292555139670898650, 2.73599594750521509939223006524, 3.06410809706778392471009430256, 3.18584403962802770143066348153, 3.24490275868073548859918395360, 3.24674561194127173003622621102

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

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