Properties

Degree 24
Conductor $ 2^{24} \cdot 5^{12} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 18·4-s − 40·8-s + 116·13-s + 156·16-s − 332·17-s + 170·25-s − 696·26-s − 680·32-s + 1.99e3·34-s + 508·37-s − 656·41-s − 1.02e3·50-s + 2.08e3·52-s − 644·53-s − 896·61-s + 2.07e3·64-s − 5.97e3·68-s + 1.43e3·73-s − 3.04e3·74-s + 2.22e3·81-s + 3.93e3·82-s − 4.77e3·97-s + 3.06e3·100-s − 2.53e3·101-s − 4.64e3·104-s + 3.86e3·106-s + ⋯
L(s)  = 1  − 2.12·2-s + 9/4·4-s − 1.76·8-s + 2.47·13-s + 2.43·16-s − 4.73·17-s + 1.35·25-s − 5.24·26-s − 3.75·32-s + 10.0·34-s + 2.25·37-s − 2.49·41-s − 2.88·50-s + 5.56·52-s − 1.66·53-s − 1.88·61-s + 4.04·64-s − 10.6·68-s + 2.30·73-s − 4.78·74-s + 3.05·81-s + 5.30·82-s − 4.99·97-s + 3.05·100-s − 2.49·101-s − 4.37·104-s + 3.54·106-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{24} \cdot 5^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{20} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(24,\ 2^{24} \cdot 5^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )$
$L(2)$  $\approx$  $0.391019$
$L(\frac12)$  $\approx$  $0.391019$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 24. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 23.
$p$$F_p$
bad2 \( 1 + 3 p T + 9 p T^{2} + 5 p^{3} T^{3} - p^{8} T^{5} - 23 p^{5} T^{6} - p^{11} T^{7} + 5 p^{12} T^{9} + 9 p^{13} T^{10} + 3 p^{16} T^{11} + p^{18} T^{12} \)
5 \( ( 1 - 17 p T^{2} - 16 p^{2} T^{3} - 17 p^{4} T^{4} + p^{9} T^{6} )^{2} \)
good3 \( 1 - 742 p T^{4} + 287575 p^{2} T^{8} - 2136402140 T^{12} + 287575 p^{14} T^{16} - 742 p^{25} T^{20} + p^{36} T^{24} \)
7 \( 1 - 163746 T^{4} - 12219131425 T^{8} + 4249089576371140 T^{12} - 12219131425 p^{12} T^{16} - 163746 p^{24} T^{20} + p^{36} T^{24} \)
11 \( ( 1 - 4986 T^{2} + 12380375 T^{4} - 19918383340 T^{6} + 12380375 p^{6} T^{8} - 4986 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
13 \( ( 1 - 58 T + 1682 T^{2} - 72450 T^{3} + 7507335 T^{4} - 469328492 T^{5} + 17218216316 T^{6} - 469328492 p^{3} T^{7} + 7507335 p^{6} T^{8} - 72450 p^{9} T^{9} + 1682 p^{12} T^{10} - 58 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
17 \( ( 1 + 166 T + 13778 T^{2} + 1478470 T^{3} + 180009375 T^{4} + 13146146484 T^{5} + 795027918044 T^{6} + 13146146484 p^{3} T^{7} + 180009375 p^{6} T^{8} + 1478470 p^{9} T^{9} + 13778 p^{12} T^{10} + 166 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
19 \( ( 1 + 16994 T^{2} + 156289815 T^{4} + 1042328090300 T^{6} + 156289815 p^{6} T^{8} + 16994 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
23 \( 1 + 86914334 T^{4} + 54372416869500575 T^{8} + \)\(33\!\cdots\!60\)\( T^{12} + 54372416869500575 p^{12} T^{16} + 86914334 p^{24} T^{20} + p^{36} T^{24} \)
29 \( ( 1 - 80686 T^{2} + 2793486135 T^{4} - 69052346800420 T^{6} + 2793486135 p^{6} T^{8} - 80686 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
31 \( ( 1 - 3326 p T^{2} + 5864634815 T^{4} - 214293224466300 T^{6} + 5864634815 p^{6} T^{8} - 3326 p^{13} T^{10} + p^{18} T^{12} )^{2} \)
37 \( ( 1 - 254 T + 32258 T^{2} + 4711770 T^{3} - 385957065 T^{4} - 961296674116 T^{5} + 267719946494684 T^{6} - 961296674116 p^{3} T^{7} - 385957065 p^{6} T^{8} + 4711770 p^{9} T^{9} + 32258 p^{12} T^{10} - 254 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
41 \( ( 1 + 4 p T + 188335 T^{2} + 20815080 T^{3} + 188335 p^{3} T^{4} + 4 p^{7} T^{5} + p^{9} T^{6} )^{4} \)
43 \( 1 + 9995711534 T^{4} - 4601157531678236945 T^{8} - \)\(23\!\cdots\!20\)\( p^{2} T^{12} - 4601157531678236945 p^{12} T^{16} + 9995711534 p^{24} T^{20} + p^{36} T^{24} \)
47 \( 1 - 14480422466 T^{4} + \)\(18\!\cdots\!35\)\( T^{8} - \)\(27\!\cdots\!80\)\( T^{12} + \)\(18\!\cdots\!35\)\( p^{12} T^{16} - 14480422466 p^{24} T^{20} + p^{36} T^{24} \)
53 \( ( 1 + 322 T + 51842 T^{2} + 24976890 T^{3} + 1487618135 T^{4} + 291824210108 T^{5} + 328768813336156 T^{6} + 291824210108 p^{3} T^{7} + 1487618135 p^{6} T^{8} + 24976890 p^{9} T^{9} + 51842 p^{12} T^{10} + 322 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
59 \( ( 1 + 543794 T^{2} + 146689289095 T^{4} + 30962597079669980 T^{6} + 146689289095 p^{6} T^{8} + 543794 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
61 \( ( 1 + 224 T + 625475 T^{2} + 89988720 T^{3} + 625475 p^{3} T^{4} + 224 p^{6} T^{5} + p^{9} T^{6} )^{4} \)
67 \( 1 + 152118288974 T^{4} - \)\(67\!\cdots\!65\)\( T^{8} - \)\(22\!\cdots\!80\)\( T^{12} - \)\(67\!\cdots\!65\)\( p^{12} T^{16} + 152118288974 p^{24} T^{20} + p^{36} T^{24} \)
71 \( ( 1 - 1431506 T^{2} + 1032027741935 T^{4} - 460759845466007580 T^{6} + 1032027741935 p^{6} T^{8} - 1431506 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
73 \( ( 1 - 718 T + 257762 T^{2} - 319815230 T^{3} + 531457295055 T^{4} - 223706294377092 T^{5} + 74772514744761596 T^{6} - 223706294377092 p^{3} T^{7} + 531457295055 p^{6} T^{8} - 319815230 p^{9} T^{9} + 257762 p^{12} T^{10} - 718 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
79 \( ( 1 + 574874 T^{2} + 335775077615 T^{4} + 287028695918457900 T^{6} + 335775077615 p^{6} T^{8} + 574874 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
83 \( 1 - 1040598079986 T^{4} + \)\(60\!\cdots\!75\)\( T^{8} - \)\(23\!\cdots\!40\)\( T^{12} + \)\(60\!\cdots\!75\)\( p^{12} T^{16} - 1040598079986 p^{24} T^{20} + p^{36} T^{24} \)
89 \( ( 1 - 2798646 T^{2} + 3790036251455 T^{4} - 3254227731213032180 T^{6} + 3790036251455 p^{6} T^{8} - 2798646 p^{12} T^{10} + p^{18} T^{12} )^{2} \)
97 \( ( 1 + 2386 T + 2846498 T^{2} + 2507112050 T^{3} + 1418732317695 T^{4} + 602523284624124 T^{5} + 542007267886579004 T^{6} + 602523284624124 p^{3} T^{7} + 1418732317695 p^{6} T^{8} + 2507112050 p^{9} T^{9} + 2846498 p^{12} T^{10} + 2386 p^{15} T^{11} + p^{18} T^{12} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.90094636540833967842366730723, −6.74045873526998573045375295989, −6.68156544128264921793062924906, −6.57999626026445557564024249492, −6.52607985161147397187458486045, −6.21998371934022356453435387675, −6.08003447560574532295963258695, −5.87755051499836557867356258851, −5.80065954854311085639031138187, −5.61241536916767869930048852112, −5.09659175642776025805495108126, −4.98389133627466619250463836693, −4.86748403966537872443889530926, −4.66690016842414864037924154838, −4.24966230655131450597625971537, −4.15265650491696262188221241899, −4.10991926619355091410625798215, −3.59623023422900153389344886929, −3.30343824573709458557378305947, −3.06566795812459389820351851678, −2.89403133439151454233574161653, −2.15758103342252538819102954414, −2.00358922228275972764457728887, −1.51891437296429275105910588701, −0.61822113697246821664982706940, 0.61822113697246821664982706940, 1.51891437296429275105910588701, 2.00358922228275972764457728887, 2.15758103342252538819102954414, 2.89403133439151454233574161653, 3.06566795812459389820351851678, 3.30343824573709458557378305947, 3.59623023422900153389344886929, 4.10991926619355091410625798215, 4.15265650491696262188221241899, 4.24966230655131450597625971537, 4.66690016842414864037924154838, 4.86748403966537872443889530926, 4.98389133627466619250463836693, 5.09659175642776025805495108126, 5.61241536916767869930048852112, 5.80065954854311085639031138187, 5.87755051499836557867356258851, 6.08003447560574532295963258695, 6.21998371934022356453435387675, 6.52607985161147397187458486045, 6.57999626026445557564024249492, 6.68156544128264921793062924906, 6.74045873526998573045375295989, 6.90094636540833967842366730723

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

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