↑

Properties

Label 24-1014e12-1.1-c1e12-0-0
Degree $24$
Conductor $1.182\times 10^{36}$
Sign $1$
Analytic cond. $7.93927\times 10^{10}$
Root an. cond. $2.84549$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 3·4-s + 15·9-s + 18·12-s + 3·16-s − 16·17-s − 6·25-s + 14·27-s − 22·29-s + 45·36-s + 24·43-s + 18·48-s − 49-s − 96·51-s + 20·53-s − 44·61-s − 2·64-s − 48·68-s − 36·75-s + 124·79-s − 21·81-s − 132·87-s − 18·100-s − 26·101-s + 4·103-s − 2·107-s + 42·108-s + ⋯
L(s)  = 1  + 3.46·3-s + 3/2·4-s + 5·9-s + 5.19·12-s + 3/4·16-s − 3.88·17-s − 6/5·25-s + 2.69·27-s − 4.08·29-s + 15/2·36-s + 3.65·43-s + 2.59·48-s − 1/7·49-s − 13.4·51-s + 2.74·53-s − 5.63·61-s − 1/4·64-s − 5.82·68-s − 4.15·75-s + 13.9·79-s − 7/3·81-s − 14.1·87-s − 9/5·100-s − 2.58·101-s + 0.394·103-s − 0.193·107-s + 4.04·108-s + ⋯

Functional equation

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

Invariants

Degree: \(24\)
Conductor: \(2^{12} \cdot 3^{12} \cdot 13^{24}\)
Sign: $1$
Analytic conductor: \(7.93927\times 10^{10}\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{12} \cdot 3^{12} \cdot 13^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.009617233681\)
\(L(\frac12)\) \(\approx\) \(0.009617233681\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T^{2} + T^{4} )^{3} \)
3 \( ( 1 - T + T^{2} )^{6} \)
13 \( 1 \)
good5 \( ( 1 + 3 T^{2} + 29 T^{4} + 151 T^{6} + 29 p^{2} T^{8} + 3 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
7 \( 1 + T^{2} - 4 T^{4} - 187 p T^{6} - 877 T^{8} + 5196 T^{10} + 663321 T^{12} + 5196 p^{2} T^{14} - 877 p^{4} T^{16} - 187 p^{7} T^{18} - 4 p^{8} T^{20} + p^{10} T^{22} + p^{12} T^{24} \)
11 \( 1 + 25 T^{2} + 560 T^{4} + 5543 T^{6} + 45335 T^{8} - 154560 T^{10} - 2281047 T^{12} - 154560 p^{2} T^{14} + 45335 p^{4} T^{16} + 5543 p^{6} T^{18} + 560 p^{8} T^{20} + 25 p^{10} T^{22} + p^{12} T^{24} \)
17 \( ( 1 + 8 T + T^{2} - 24 T^{3} + 786 T^{4} + 1808 T^{5} - 6447 T^{6} + 1808 p T^{7} + 786 p^{2} T^{8} - 24 p^{3} T^{9} + p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
19 \( 1 + 34 T^{2} + 15 p T^{4} - 6730 T^{6} - 173638 T^{8} - 1122582 T^{10} - 207131 T^{12} - 1122582 p^{2} T^{14} - 173638 p^{4} T^{16} - 6730 p^{6} T^{18} + 15 p^{9} T^{20} + 34 p^{10} T^{22} + p^{12} T^{24} \)
23 \( ( 1 - 41 T^{2} - 112 T^{3} + 738 T^{4} + 2296 T^{5} - 11193 T^{6} + 2296 p T^{7} + 738 p^{2} T^{8} - 112 p^{3} T^{9} - 41 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
29 \( ( 1 + 11 T + 10 T^{2} + 3 T^{3} + 2403 T^{4} + 208 p T^{5} - 1359 p T^{6} + 208 p^{2} T^{7} + 2403 p^{2} T^{8} + 3 p^{3} T^{9} + 10 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
31 \( ( 1 - 61 T^{2} + 2665 T^{4} - 91945 T^{6} + 2665 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
37 \( 1 + 70 T^{2} + 4797 T^{4} + 172802 T^{6} + 5665322 T^{8} + 102922302 T^{10} + 2788717813 T^{12} + 102922302 p^{2} T^{14} + 5665322 p^{4} T^{16} + 172802 p^{6} T^{18} + 4797 p^{8} T^{20} + 70 p^{10} T^{22} + p^{12} T^{24} \)
41 \( 1 + 114 T^{2} + 4405 T^{4} + 168950 T^{6} + 13169642 T^{8} + 510657658 T^{10} + 12906706109 T^{12} + 510657658 p^{2} T^{14} + 13169642 p^{4} T^{16} + 168950 p^{6} T^{18} + 4405 p^{8} T^{20} + 114 p^{10} T^{22} + p^{12} T^{24} \)
43 \( ( 1 - 12 T - 5 T^{2} + 68 T^{3} + 4658 T^{4} - 6692 T^{5} - 200701 T^{6} - 6692 p T^{7} + 4658 p^{2} T^{8} + 68 p^{3} T^{9} - 5 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
47 \( ( 1 - 202 T^{2} + 19631 T^{4} - 1156428 T^{6} + 19631 p^{2} T^{8} - 202 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
53 \( ( 1 - 5 T + 123 T^{2} - 573 T^{3} + 123 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
59 \( 1 + 257 T^{2} + 33760 T^{4} + 3490799 T^{6} + 312421103 T^{8} + 22955849144 T^{10} + 1432753743161 T^{12} + 22955849144 p^{2} T^{14} + 312421103 p^{4} T^{16} + 3490799 p^{6} T^{18} + 33760 p^{8} T^{20} + 257 p^{10} T^{22} + p^{12} T^{24} \)
61 \( ( 1 + 22 T + 149 T^{2} + 1346 T^{3} + 25526 T^{4} + 191742 T^{5} + 879417 T^{6} + 191742 p T^{7} + 25526 p^{2} T^{8} + 1346 p^{3} T^{9} + 149 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \)
67 \( 1 + 334 T^{2} + 60941 T^{4} + 8156474 T^{6} + 13038734 p T^{8} + 76367401254 T^{10} + 5568637241061 T^{12} + 76367401254 p^{2} T^{14} + 13038734 p^{5} T^{16} + 8156474 p^{6} T^{18} + 60941 p^{8} T^{20} + 334 p^{10} T^{22} + p^{12} T^{24} \)
71 \( 1 + 262 T^{2} + 32917 T^{4} + 3065458 T^{6} + 271671626 T^{8} + 24736579870 T^{10} + 1984221942653 T^{12} + 24736579870 p^{2} T^{14} + 271671626 p^{4} T^{16} + 3065458 p^{6} T^{18} + 32917 p^{8} T^{20} + 262 p^{10} T^{22} + p^{12} T^{24} \)
73 \( ( 1 - 209 T^{2} + 14305 T^{4} - 638873 T^{6} + 14305 p^{2} T^{8} - 209 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
79 \( ( 1 - 31 T + 513 T^{2} - 5431 T^{3} + 513 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} )^{4} \)
83 \( ( 1 - 73 T^{2} + 19361 T^{4} - 896217 T^{6} + 19361 p^{2} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} )^{2} \)
89 \( 1 + 226 T^{2} + 37205 T^{4} + 2543750 T^{6} + 15738602 T^{8} - 31382937558 T^{10} - 3647348660451 T^{12} - 31382937558 p^{2} T^{14} + 15738602 p^{4} T^{16} + 2543750 p^{6} T^{18} + 37205 p^{8} T^{20} + 226 p^{10} T^{22} + p^{12} T^{24} \)
97 \( 1 + 233 T^{2} + 44928 T^{4} + 3686359 T^{6} + 174911831 T^{8} - 283115112 p T^{10} - 361172879 p^{2} T^{12} - 283115112 p^{3} T^{14} + 174911831 p^{4} T^{16} + 3686359 p^{6} T^{18} + 44928 p^{8} T^{20} + 233 p^{10} T^{22} + p^{12} T^{24} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.24930830927308384512608542227, −2.97833382222493824503443059479, −2.93777996934053882708910019967, −2.86188016913223535896736737967, −2.65952403619269291300121019298, −2.53837715967488905510546367081, −2.47726315999522166539472169093, −2.44046195055215531370009927114, −2.42184522892591391666355277332, −2.21895523008014369296045943954, −2.19360292997510324133727918395, −2.16818690368002059601434247587, −2.16016375591529533913084420915, −2.10920986155009568641248323977, −1.99713997687850451049693022598, −1.74002214049556484415371017664, −1.45796610040519098674853237700, −1.42373208103121375824904366364, −1.41764159237509913736777766700, −1.23123882533038882980903292049, −1.06774341750737866279433902259, −0.75052185771599464150225284808, −0.70315360204804764395148771796, −0.092556461876884014049368779289, −0.01406836951807021542236876520, 0.01406836951807021542236876520, 0.092556461876884014049368779289, 0.70315360204804764395148771796, 0.75052185771599464150225284808, 1.06774341750737866279433902259, 1.23123882533038882980903292049, 1.41764159237509913736777766700, 1.42373208103121375824904366364, 1.45796610040519098674853237700, 1.74002214049556484415371017664, 1.99713997687850451049693022598, 2.10920986155009568641248323977, 2.16016375591529533913084420915, 2.16818690368002059601434247587, 2.19360292997510324133727918395, 2.21895523008014369296045943954, 2.42184522892591391666355277332, 2.44046195055215531370009927114, 2.47726315999522166539472169093, 2.53837715967488905510546367081, 2.65952403619269291300121019298, 2.86188016913223535896736737967, 2.93777996934053882708910019967, 2.97833382222493824503443059479, 3.24930830927308384512608542227

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

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