Properties

Label 24-475e12-1.1-c1e12-0-5
Degree $24$
Conductor $1.319\times 10^{32}$
Sign $1$
Analytic cond. $8.86438\times 10^{6}$
Root an. cond. $1.94753$
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  + 5·4-s + 5·9-s + 4·11-s + 10·16-s + 12·19-s + 12·29-s + 60·31-s + 25·36-s − 12·41-s + 20·44-s − 40·49-s − 20·59-s + 2·61-s − 5·64-s + 2·71-s + 60·76-s − 24·79-s + 15·81-s − 36·89-s + 20·99-s − 14·101-s + 50·109-s + 60·116-s − 106·121-s + 300·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 5/2·4-s + 5/3·9-s + 1.20·11-s + 5/2·16-s + 2.75·19-s + 2.22·29-s + 10.7·31-s + 25/6·36-s − 1.87·41-s + 3.01·44-s − 5.71·49-s − 2.60·59-s + 0.256·61-s − 5/8·64-s + 0.237·71-s + 6.88·76-s − 2.70·79-s + 5/3·81-s − 3.81·89-s + 2.01·99-s − 1.39·101-s + 4.78·109-s + 5.57·116-s − 9.63·121-s + 26.9·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(5^{24} \cdot 19^{12}\)
Sign: $1$
Analytic conductor: \(8.86438\times 10^{6}\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 5^{24} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(24.41907978\)
\(L(\frac12)\) \(\approx\) \(24.41907978\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( ( 1 - 6 T + 30 T^{2} - 115 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
good2 \( 1 - 5 T^{2} + 15 T^{4} - 5 p^{2} T^{6} - 15 T^{8} + 85 p T^{10} - 447 T^{12} + 85 p^{3} T^{14} - 15 p^{4} T^{16} - 5 p^{8} T^{18} + 15 p^{8} T^{20} - 5 p^{10} T^{22} + p^{12} T^{24} \)
3 \( 1 - 5 T^{2} + 10 T^{4} - 17 T^{6} - 55 T^{8} + 170 p T^{10} - 1751 T^{12} + 170 p^{3} T^{14} - 55 p^{4} T^{16} - 17 p^{6} T^{18} + 10 p^{8} T^{20} - 5 p^{10} T^{22} + p^{12} T^{24} \)
7 \( ( 1 + 20 T^{2} + 22 p T^{4} + 873 T^{6} + 22 p^{3} T^{8} + 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
11 \( ( 1 - T + 29 T^{2} - 19 T^{3} + 29 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{4} \)
13 \( 1 - 47 T^{2} + 1047 T^{4} - 17192 T^{6} + 274449 T^{8} - 4625017 T^{10} + 68472654 T^{12} - 4625017 p^{2} T^{14} + 274449 p^{4} T^{16} - 17192 p^{6} T^{18} + 1047 p^{8} T^{20} - 47 p^{10} T^{22} + p^{12} T^{24} \)
17 \( 1 - 67 T^{2} + 2336 T^{4} - 55809 T^{6} + 1050385 T^{8} - 17426642 T^{10} + 289213009 T^{12} - 17426642 p^{2} T^{14} + 1050385 p^{4} T^{16} - 55809 p^{6} T^{18} + 2336 p^{8} T^{20} - 67 p^{10} T^{22} + p^{12} T^{24} \)
23 \( 1 - 126 T^{2} + 9038 T^{4} - 450442 T^{6} + 17238744 T^{8} - 529408036 T^{10} + 13378463315 T^{12} - 529408036 p^{2} T^{14} + 17238744 p^{4} T^{16} - 450442 p^{6} T^{18} + 9038 p^{8} T^{20} - 126 p^{10} T^{22} + p^{12} T^{24} \)
29 \( ( 1 - 6 T - 24 T^{2} + 282 T^{3} - 66 T^{4} - 3426 T^{5} + 17143 T^{6} - 3426 p T^{7} - 66 p^{2} T^{8} + 282 p^{3} T^{9} - 24 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 15 T + 163 T^{2} - 1027 T^{3} + 163 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
37 \( ( 1 + 124 T^{2} + 6918 T^{4} + 272997 T^{6} + 6918 p^{2} T^{8} + 124 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
41 \( ( 1 + 6 T - 46 T^{2} - 486 T^{3} + 428 T^{4} + 240 p T^{5} + 40763 T^{6} + 240 p^{2} T^{7} + 428 p^{2} T^{8} - 486 p^{3} T^{9} - 46 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
43 \( 1 - 131 T^{2} + 8754 T^{4} - 307043 T^{6} + 3108159 T^{8} + 286076576 T^{10} - 18992494095 T^{12} + 286076576 p^{2} T^{14} + 3108159 p^{4} T^{16} - 307043 p^{6} T^{18} + 8754 p^{8} T^{20} - 131 p^{10} T^{22} + p^{12} T^{24} \)
47 \( 1 - 68 T^{2} - 1686 T^{4} + 135466 T^{6} + 6682746 T^{8} - 217285498 T^{10} - 9280932657 T^{12} - 217285498 p^{2} T^{14} + 6682746 p^{4} T^{16} + 135466 p^{6} T^{18} - 1686 p^{8} T^{20} - 68 p^{10} T^{22} + p^{12} T^{24} \)
53 \( 1 - 219 T^{2} + 24512 T^{4} - 2052217 T^{6} + 146386377 T^{8} - 9025695874 T^{10} + 497746964057 T^{12} - 9025695874 p^{2} T^{14} + 146386377 p^{4} T^{16} - 2052217 p^{6} T^{18} + 24512 p^{8} T^{20} - 219 p^{10} T^{22} + p^{12} T^{24} \)
59 \( ( 1 + 10 T - 22 T^{2} - 558 T^{3} - 1204 T^{4} - 692 T^{5} - 25177 T^{6} - 692 p T^{7} - 1204 p^{2} T^{8} - 558 p^{3} T^{9} - 22 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
61 \( ( 1 - T - 141 T^{2} - 124 T^{3} + 11493 T^{4} + 12325 T^{5} - 779682 T^{6} + 12325 p T^{7} + 11493 p^{2} T^{8} - 124 p^{3} T^{9} - 141 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 - 326 T^{2} + 57453 T^{4} - 7511138 T^{6} + 794477802 T^{8} - 68855305342 T^{10} + 5000651443461 T^{12} - 68855305342 p^{2} T^{14} + 794477802 p^{4} T^{16} - 7511138 p^{6} T^{18} + 57453 p^{8} T^{20} - 326 p^{10} T^{22} + p^{12} T^{24} \)
71 \( ( 1 - T - 88 T^{2} + 1149 T^{3} + 983 T^{4} - 47494 T^{5} + 531551 T^{6} - 47494 p T^{7} + 983 p^{2} T^{8} + 1149 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \)
73 \( 1 - 202 T^{2} + 16450 T^{4} - 756354 T^{6} + 42856296 T^{8} - 3999054676 T^{10} + 316147256779 T^{12} - 3999054676 p^{2} T^{14} + 42856296 p^{4} T^{16} - 756354 p^{6} T^{18} + 16450 p^{8} T^{20} - 202 p^{10} T^{22} + p^{12} T^{24} \)
79 \( ( 1 + 12 T - 61 T^{2} - 436 T^{3} + 8454 T^{4} - 16948 T^{5} - 1135897 T^{6} - 16948 p T^{7} + 8454 p^{2} T^{8} - 436 p^{3} T^{9} - 61 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
83 \( ( 1 + 39 T^{2} + 7249 T^{4} + 843277 T^{6} + 7249 p^{2} T^{8} + 39 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( ( 1 + 18 T + 29 T^{2} - 954 T^{3} + 4394 T^{4} + 121338 T^{5} + 907733 T^{6} + 121338 p T^{7} + 4394 p^{2} T^{8} - 954 p^{3} T^{9} + 29 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
97 \( 1 - 53 T^{2} - 7266 T^{4} + 2553571 T^{6} - 75110019 T^{8} - 10197223318 T^{10} + 2845433909433 T^{12} - 10197223318 p^{2} T^{14} - 75110019 p^{4} T^{16} + 2553571 p^{6} T^{18} - 7266 p^{8} T^{20} - 53 p^{10} T^{22} + p^{12} T^{24} \)
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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

−3.61101857869628465214667276316, −3.27731147246231118947269076682, −3.26735051691760096730612472332, −3.21669337015762043844747614566, −3.13637581438554406600844331283, −3.11495598124505707662667730001, −3.09929594941308009529118919225, −2.87743556853289236613669667394, −2.78721755169762690654306766434, −2.73902262646111318230320050660, −2.64116635791619730581787046315, −2.57266278393050319686771701531, −2.40906861509876142706981586935, −2.27641312930943371682793972174, −1.91081745322542058181648413497, −1.81509914330790253450636612623, −1.70252752471780801722845134413, −1.68762232768748586571225411580, −1.47173443129065245380355518329, −1.35524937247830206127772435533, −1.08170690215184257773865194515, −1.03881526281933039060097398099, −1.03814894808621749359773214823, −0.934105710796841451014442690212, −0.32245129511521123450540693935, 0.32245129511521123450540693935, 0.934105710796841451014442690212, 1.03814894808621749359773214823, 1.03881526281933039060097398099, 1.08170690215184257773865194515, 1.35524937247830206127772435533, 1.47173443129065245380355518329, 1.68762232768748586571225411580, 1.70252752471780801722845134413, 1.81509914330790253450636612623, 1.91081745322542058181648413497, 2.27641312930943371682793972174, 2.40906861509876142706981586935, 2.57266278393050319686771701531, 2.64116635791619730581787046315, 2.73902262646111318230320050660, 2.78721755169762690654306766434, 2.87743556853289236613669667394, 3.09929594941308009529118919225, 3.11495598124505707662667730001, 3.13637581438554406600844331283, 3.21669337015762043844747614566, 3.26735051691760096730612472332, 3.27731147246231118947269076682, 3.61101857869628465214667276316

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

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