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Properties

Label 24-735e12-1.1-c1e12-0-0
Degree $24$
Conductor $2.486\times 10^{34}$
Sign $1$
Analytic cond. $1.67021\times 10^{9}$
Root an. cond. $2.42260$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·5-s + 3·9-s − 12·11-s − 16-s + 12·19-s + 2·20-s + 3·25-s − 8·29-s − 4·31-s − 3·36-s + 8·41-s + 12·44-s − 6·45-s + 24·55-s + 32·59-s + 12·61-s + 10·64-s + 24·71-s − 12·76-s + 24·79-s + 2·80-s + 3·81-s + 28·89-s − 24·95-s − 36·99-s − 3·100-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.894·5-s + 9-s − 3.61·11-s − 1/4·16-s + 2.75·19-s + 0.447·20-s + 3/5·25-s − 1.48·29-s − 0.718·31-s − 1/2·36-s + 1.24·41-s + 1.80·44-s − 0.894·45-s + 3.23·55-s + 4.16·59-s + 1.53·61-s + 5/4·64-s + 2.84·71-s − 1.37·76-s + 2.70·79-s + 0.223·80-s + 1/3·81-s + 2.96·89-s − 2.46·95-s − 3.61·99-s − 0.299·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(3^{12} \cdot 5^{12} \cdot 7^{24}\)
Sign: $1$
Analytic conductor: \(1.67021\times 10^{9}\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 3^{12} \cdot 5^{12} \cdot 7^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.516823614\)
\(L(\frac12)\) \(\approx\) \(2.516823614\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + T^{2} - 18 T^{3} - 6 p T^{4} + 26 T^{5} + 249 T^{6} + 26 p T^{7} - 6 p^{3} T^{8} - 18 p^{3} T^{9} + p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 \)
good2 \( 1 + T^{2} + p T^{4} - 7 T^{6} + p T^{8} + 21 T^{10} + 117 T^{12} + 21 p^{2} T^{14} + p^{5} T^{16} - 7 p^{6} T^{18} + p^{9} T^{20} + p^{10} T^{22} + p^{12} T^{24} \)
11 \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{6} \)
13 \( ( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
17 \( 1 + 70 T^{2} + 2485 T^{4} + 66610 T^{6} + 1548890 T^{8} + 31831870 T^{10} + 576997613 T^{12} + 31831870 p^{2} T^{14} + 1548890 p^{4} T^{16} + 66610 p^{6} T^{18} + 2485 p^{8} T^{20} + 70 p^{10} T^{22} + p^{12} T^{24} \)
19 \( ( 1 - 6 T - 17 T^{2} + 58 T^{3} + 674 T^{4} - 46 T^{5} - 17305 T^{6} - 46 p T^{7} + 674 p^{2} T^{8} + 58 p^{3} T^{9} - 17 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
23 \( 1 + 106 T^{2} + 6053 T^{4} + 248702 T^{6} + 8184794 T^{8} + 227653746 T^{10} + 5545513725 T^{12} + 227653746 p^{2} T^{14} + 8184794 p^{4} T^{16} + 248702 p^{6} T^{18} + 6053 p^{8} T^{20} + 106 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
31 \( ( 1 + 2 T - 37 T^{2} + 202 T^{3} + 530 T^{4} - 5622 T^{5} + 7227 T^{6} - 5622 p T^{7} + 530 p^{2} T^{8} + 202 p^{3} T^{9} - 37 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
37 \( 1 + 46 T^{2} + 717 T^{4} - 85222 T^{6} - 3398278 T^{8} + 14638566 T^{10} + 4079564917 T^{12} + 14638566 p^{2} T^{14} - 3398278 p^{4} T^{16} - 85222 p^{6} T^{18} + 717 p^{8} T^{20} + 46 p^{10} T^{22} + p^{12} T^{24} \)
41 \( ( 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
43 \( ( 1 + 46 T^{2} + 2839 T^{4} + 118948 T^{6} + 2839 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
47 \( 1 + 154 T^{2} + 11573 T^{4} + 565358 T^{6} + 20169434 T^{8} + 410981154 T^{10} + 5646494445 T^{12} + 410981154 p^{2} T^{14} + 20169434 p^{4} T^{16} + 565358 p^{6} T^{18} + 11573 p^{8} T^{20} + 154 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 + 146 T^{2} + 7213 T^{4} + 289382 T^{6} + 30094394 T^{8} + 2063064026 T^{10} + 100360862165 T^{12} + 2063064026 p^{2} T^{14} + 30094394 p^{4} T^{16} + 289382 p^{6} T^{18} + 7213 p^{8} T^{20} + 146 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 - 16 T + 143 T^{2} - 592 T^{3} - 3626 T^{4} + 88944 T^{5} - 864085 T^{6} + 88944 p T^{7} - 3626 p^{2} T^{8} - 592 p^{3} T^{9} + 143 p^{4} T^{10} - 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - 6 T - 95 T^{2} + 182 T^{3} + 6266 T^{4} + 10162 T^{5} - 492559 T^{6} + 10162 p T^{7} + 6266 p^{2} T^{8} + 182 p^{3} T^{9} - 95 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 274 T^{2} + 38973 T^{4} + 3966278 T^{6} + 329505914 T^{8} + 23681216154 T^{10} + 1596172229125 T^{12} + 23681216154 p^{2} T^{14} + 329505914 p^{4} T^{16} + 3966278 p^{6} T^{18} + 38973 p^{8} T^{20} + 274 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - 2 T + p T^{2} )^{12} \)
73 \( 1 + 298 T^{2} + 45029 T^{4} + 5080382 T^{6} + 496253018 T^{8} + 43754080818 T^{10} + 3427677334845 T^{12} + 43754080818 p^{2} T^{14} + 496253018 p^{4} T^{16} + 5080382 p^{6} T^{18} + 45029 p^{8} T^{20} + 298 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 - 12 T - 77 T^{2} + 500 T^{3} + 12470 T^{4} - 4172 T^{5} - 1287289 T^{6} - 4172 p T^{7} + 12470 p^{2} T^{8} + 500 p^{3} T^{9} - 77 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 - 14 T - 123 T^{2} + 598 T^{3} + 37922 T^{4} - 123654 T^{5} - 2788283 T^{6} - 123654 p T^{7} + 37922 p^{2} T^{8} + 598 p^{3} T^{9} - 123 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( ( 1 - 26 T^{2} + 8719 T^{4} + 446932 T^{6} + 8719 p^{2} T^{8} - 26 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
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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

−3.38173997409051537843459692786, −3.36517708343086212298136704385, −3.14158793766893021013526133102, −3.07055579015302694986961295305, −3.00013456199197483998078757076, −2.82659157255741781947265176840, −2.80271829263982882897193761760, −2.72397167590472067478437314646, −2.46856193675463577125340973753, −2.40074047144575685416739328759, −2.31658493553249715616327039469, −2.22980091733756435314805183149, −2.18666793887950695988208025001, −2.03140870983165599720888098168, −1.84022625232667932833102178031, −1.78939770471450205370702708364, −1.66073312672501879422302646716, −1.61983298445998794527616068455, −1.05925965489173256460656674755, −0.968926579001728051071307376238, −0.875693076560454965703048661132, −0.850168046451461651510580526547, −0.67875154723938648913771673678, −0.49944049372602922030395339116, −0.17798544785847505235991490465, 0.17798544785847505235991490465, 0.49944049372602922030395339116, 0.67875154723938648913771673678, 0.850168046451461651510580526547, 0.875693076560454965703048661132, 0.968926579001728051071307376238, 1.05925965489173256460656674755, 1.61983298445998794527616068455, 1.66073312672501879422302646716, 1.78939770471450205370702708364, 1.84022625232667932833102178031, 2.03140870983165599720888098168, 2.18666793887950695988208025001, 2.22980091733756435314805183149, 2.31658493553249715616327039469, 2.40074047144575685416739328759, 2.46856193675463577125340973753, 2.72397167590472067478437314646, 2.80271829263982882897193761760, 2.82659157255741781947265176840, 3.00013456199197483998078757076, 3.07055579015302694986961295305, 3.14158793766893021013526133102, 3.36517708343086212298136704385, 3.38173997409051537843459692786

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

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