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Properties

Label 24-24e24-1.1-c7e12-0-1
Degree $24$
Conductor $1.334\times 10^{33}$
Sign $1$
Analytic cond. $1.15173\times 10^{27}$
Root an. cond. $13.4139$
Motivic weight $7$
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  − 4.87e3·17-s + 2.47e5·25-s − 3.56e5·41-s − 1.71e6·49-s + 2.68e7·73-s − 4.53e7·89-s + 8.24e7·97-s − 1.91e7·113-s + 9.79e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.06e8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 0.240·17-s + 3.17·25-s − 0.808·41-s − 2.08·49-s + 8.06·73-s − 6.82·89-s + 9.16·97-s − 1.24·113-s + 5.02·121-s + 4.87·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{72} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.15173\times 10^{27}\)
Root analytic conductor: \(13.4139\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{72} \cdot 3^{24} ,\ ( \ : [7/2]^{12} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(26.63202066\)
\(L(\frac12)\) \(\approx\) \(26.63202066\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - 123882 T^{2} + 120079059 p^{3} T^{4} - 2157102667372 p^{4} T^{6} + 120079059 p^{17} T^{8} - 123882 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
7 \( ( 1 + 122730 p T^{2} + 526171371135 T^{4} + 750853771700795156 T^{6} + 526171371135 p^{14} T^{8} + 122730 p^{29} T^{10} + p^{42} T^{12} )^{2} \)
11 \( ( 1 - 4452342 p T^{2} + 1034007392287239 T^{4} - \)\(17\!\cdots\!76\)\( T^{6} + 1034007392287239 p^{14} T^{8} - 4452342 p^{29} T^{10} + p^{42} T^{12} )^{2} \)
13 \( ( 1 - 153004062 T^{2} + 16626793890728631 T^{4} - \)\(11\!\cdots\!44\)\( T^{6} + 16626793890728631 p^{14} T^{8} - 153004062 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
17 \( ( 1 + 1218 T - 154523649 T^{2} - 3049988326692 T^{3} - 154523649 p^{7} T^{4} + 1218 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
19 \( ( 1 - 632327346 T^{2} + 642867890301313335 T^{4} - \)\(15\!\cdots\!00\)\( T^{6} + 642867890301313335 p^{14} T^{8} - 632327346 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
23 \( ( 1 + 15655954650 T^{2} + \)\(11\!\cdots\!19\)\( T^{4} + \)\(50\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!19\)\( p^{14} T^{8} + 15655954650 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
29 \( ( 1 - 79902346074 T^{2} + \)\(28\!\cdots\!31\)\( T^{4} - \)\(61\!\cdots\!28\)\( T^{6} + \)\(28\!\cdots\!31\)\( p^{14} T^{8} - 79902346074 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
31 \( ( 1 + 118460811798 T^{2} + \)\(63\!\cdots\!99\)\( T^{4} + \)\(21\!\cdots\!56\)\( T^{6} + \)\(63\!\cdots\!99\)\( p^{14} T^{8} + 118460811798 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
37 \( ( 1 - 283477875774 T^{2} + \)\(53\!\cdots\!11\)\( T^{4} - \)\(59\!\cdots\!28\)\( T^{6} + \)\(53\!\cdots\!11\)\( p^{14} T^{8} - 283477875774 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
41 \( ( 1 + 89166 T + 295028411127 T^{2} + 76956645349790436 T^{3} + 295028411127 p^{7} T^{4} + 89166 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
43 \( ( 1 - 863787140514 T^{2} + \)\(44\!\cdots\!47\)\( T^{4} - \)\(14\!\cdots\!72\)\( T^{6} + \)\(44\!\cdots\!47\)\( p^{14} T^{8} - 863787140514 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
47 \( ( 1 + 1194574793802 T^{2} + \)\(12\!\cdots\!75\)\( T^{4} + \)\(67\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!75\)\( p^{14} T^{8} + 1194574793802 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
53 \( ( 1 - 4037750154762 T^{2} + \)\(91\!\cdots\!71\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{6} + \)\(91\!\cdots\!71\)\( p^{14} T^{8} - 4037750154762 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
59 \( ( 1 - 11511476319810 T^{2} + \)\(60\!\cdots\!15\)\( T^{4} - \)\(18\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!15\)\( p^{14} T^{8} - 11511476319810 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
61 \( ( 1 - 5085665437614 T^{2} + \)\(26\!\cdots\!23\)\( T^{4} - \)\(71\!\cdots\!48\)\( T^{6} + \)\(26\!\cdots\!23\)\( p^{14} T^{8} - 5085665437614 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
67 \( ( 1 + 145375820526 T^{2} + \)\(35\!\cdots\!51\)\( T^{4} + \)\(22\!\cdots\!68\)\( T^{6} + \)\(35\!\cdots\!51\)\( p^{14} T^{8} + 145375820526 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
71 \( ( 1 + 46368788910522 T^{2} + \)\(96\!\cdots\!71\)\( T^{4} + \)\(11\!\cdots\!88\)\( T^{6} + \)\(96\!\cdots\!71\)\( p^{14} T^{8} + 46368788910522 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
73 \( ( 1 - 6701718 T + 46391722698999 T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + 46391722698999 p^{7} T^{4} - 6701718 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
79 \( ( 1 + 65675172650070 T^{2} + \)\(17\!\cdots\!43\)\( T^{4} + \)\(32\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!43\)\( p^{14} T^{8} + 65675172650070 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
83 \( ( 1 - 84859614630498 T^{2} + \)\(38\!\cdots\!23\)\( T^{4} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(38\!\cdots\!23\)\( p^{14} T^{8} - 84859614630498 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
89 \( ( 1 + 11341890 T + 170598897538215 T^{2} + \)\(10\!\cdots\!96\)\( T^{3} + 170598897538215 p^{7} T^{4} + 11341890 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
97 \( ( 1 - 20603034 T + 299241578225007 T^{2} - \)\(32\!\cdots\!84\)\( T^{3} + 299241578225007 p^{7} T^{4} - 20603034 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
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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

−2.25549964647170401430095172105, −2.22860468727704385052209238701, −2.19328758610328946641743646922, −2.08175613101559120542345311672, −2.04064409265161125785957111592, −2.02894278753469624971798080444, −1.75270128246637783877640701644, −1.74365341756977607584059136160, −1.64240438564777240131415230498, −1.64227439294384081540314753413, −1.53276900655756717960566008886, −1.47084024574486420207225345787, −1.15925067057070802357125816594, −1.07082410937150301659377048635, −0.898604022054421067564769193062, −0.838841844697519427282070151117, −0.836600460032634807611414999593, −0.831501869817909633086135840820, −0.71359686261203762191528403363, −0.54311467444249877520827914693, −0.45284233862904513433611331134, −0.41724412769319920505116604328, −0.35300648547066692696818420180, −0.31451073938951921209403034546, −0.06133544749173506697674988259, 0.06133544749173506697674988259, 0.31451073938951921209403034546, 0.35300648547066692696818420180, 0.41724412769319920505116604328, 0.45284233862904513433611331134, 0.54311467444249877520827914693, 0.71359686261203762191528403363, 0.831501869817909633086135840820, 0.836600460032634807611414999593, 0.838841844697519427282070151117, 0.898604022054421067564769193062, 1.07082410937150301659377048635, 1.15925067057070802357125816594, 1.47084024574486420207225345787, 1.53276900655756717960566008886, 1.64227439294384081540314753413, 1.64240438564777240131415230498, 1.74365341756977607584059136160, 1.75270128246637783877640701644, 2.02894278753469624971798080444, 2.04064409265161125785957111592, 2.08175613101559120542345311672, 2.19328758610328946641743646922, 2.22860468727704385052209238701, 2.25549964647170401430095172105

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

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