Properties

Degree 24
Conductor $ 2^{60} \cdot 7^{12} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 25·9-s − 30·11-s + 30·17-s − 78·19-s − 121·25-s + 62·27-s − 180·33-s − 232·41-s + 200·43-s + 186·49-s + 180·51-s − 468·57-s + 110·59-s − 434·67-s + 102·73-s − 726·75-s + 178·81-s + 536·83-s + 214·89-s − 152·97-s − 750·99-s − 102·107-s − 680·113-s + 933·121-s − 1.39e3·123-s + 127-s + ⋯
L(s)  = 1  + 2·3-s + 25/9·9-s − 2.72·11-s + 1.76·17-s − 4.10·19-s − 4.83·25-s + 2.29·27-s − 5.45·33-s − 5.65·41-s + 4.65·43-s + 3.79·49-s + 3.52·51-s − 8.21·57-s + 1.86·59-s − 6.47·67-s + 1.39·73-s − 9.67·75-s + 2.19·81-s + 6.45·83-s + 2.40·89-s − 1.56·97-s − 7.57·99-s − 0.953·107-s − 6.01·113-s + 7.71·121-s − 11.3·123-s + 0.00787·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(24\)
\( N \)  =  \(2^{60} \cdot 7^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{224} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((24,\ 2^{60} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.53138\)
\(L(\frac12)\)  \(\approx\)  \(1.53138\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 24. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 23.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 186 T^{2} + 2361 p T^{4} - 2788 p^{3} T^{6} + 2361 p^{5} T^{8} - 186 p^{8} T^{10} + p^{12} T^{12} \)
good3 \( ( 1 - p T + T^{2} + 14 T^{3} - 65 T^{4} + 37 T^{5} + 514 T^{6} + 37 p^{2} T^{7} - 65 p^{4} T^{8} + 14 p^{6} T^{9} + p^{8} T^{10} - p^{11} T^{11} + p^{12} T^{12} )^{2} \)
5 \( 1 + 121 T^{2} + 7979 T^{4} + 380588 T^{6} + 14474897 T^{8} + 457025259 T^{10} + 12295779174 T^{12} + 457025259 p^{4} T^{14} + 14474897 p^{8} T^{16} + 380588 p^{12} T^{18} + 7979 p^{16} T^{20} + 121 p^{20} T^{22} + p^{24} T^{24} \)
11 \( ( 1 + 15 T - 129 T^{2} - 84 p T^{3} + 35727 T^{4} + 38013 T^{5} - 5198650 T^{6} + 38013 p^{2} T^{7} + 35727 p^{4} T^{8} - 84 p^{7} T^{9} - 129 p^{8} T^{10} + 15 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
13 \( ( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 60895 p^{4} T^{8} - 86 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
17 \( ( 1 - 15 T - 665 T^{2} + 3920 T^{3} + 23845 p T^{4} - 1221665 T^{5} - 124870762 T^{6} - 1221665 p^{2} T^{7} + 23845 p^{5} T^{8} + 3920 p^{6} T^{9} - 665 p^{8} T^{10} - 15 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
19 \( ( 1 + 39 T + 151 T^{2} - 3964 T^{3} + 263147 T^{4} + 5820949 T^{5} + 41830430 T^{6} + 5820949 p^{2} T^{7} + 263147 p^{4} T^{8} - 3964 p^{6} T^{9} + 151 p^{8} T^{10} + 39 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
23 \( 1 + 693 T^{2} - 15581 T^{4} - 315659500 T^{6} - 121343121715 T^{8} + 25399919636503 T^{10} + 46081428374834798 T^{12} + 25399919636503 p^{4} T^{14} - 121343121715 p^{8} T^{16} - 315659500 p^{12} T^{18} - 15581 p^{16} T^{20} + 693 p^{20} T^{22} + p^{24} T^{24} \)
29 \( ( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 6471151 p^{4} T^{8} - 3662 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
31 \( 1 + 3561 T^{2} + 5838879 T^{4} + 8280992264 T^{6} + 11856779149293 T^{8} + 13054196823637455 T^{10} + 12193464685853753718 T^{12} + 13054196823637455 p^{4} T^{14} + 11856779149293 p^{8} T^{16} + 8280992264 p^{12} T^{18} + 5838879 p^{16} T^{20} + 3561 p^{20} T^{22} + p^{24} T^{24} \)
37 \( 1 + 5785 T^{2} + 17889827 T^{4} + 38047414052 T^{6} + 62842022068961 T^{8} + 88030006321881747 T^{10} + \)\(11\!\cdots\!14\)\( T^{12} + 88030006321881747 p^{4} T^{14} + 62842022068961 p^{8} T^{16} + 38047414052 p^{12} T^{18} + 17889827 p^{16} T^{20} + 5785 p^{20} T^{22} + p^{24} T^{24} \)
41 \( ( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 3139 p^{2} T^{4} + 58 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
43 \( ( 1 - 50 T + 4047 T^{2} - 107900 T^{3} + 4047 p^{2} T^{4} - 50 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
47 \( 1 + 3905 T^{2} + 6537087 T^{4} + 2079696952 T^{6} - 28977286497379 T^{8} - 99900087904130553 T^{10} - \)\(24\!\cdots\!46\)\( T^{12} - 99900087904130553 p^{4} T^{14} - 28977286497379 p^{8} T^{16} + 2079696952 p^{12} T^{18} + 6537087 p^{16} T^{20} + 3905 p^{20} T^{22} + p^{24} T^{24} \)
53 \( 1 + 10561 T^{2} + 59733731 T^{4} + 209884512884 T^{6} + 494098524300977 T^{8} + 757507316642438811 T^{10} + \)\(12\!\cdots\!10\)\( T^{12} + 757507316642438811 p^{4} T^{14} + 494098524300977 p^{8} T^{16} + 209884512884 p^{12} T^{18} + 59733731 p^{16} T^{20} + 10561 p^{20} T^{22} + p^{24} T^{24} \)
59 \( ( 1 - 55 T - 7367 T^{2} + 181142 T^{3} + 51119807 T^{4} - 598293727 T^{5} - 186579818926 T^{6} - 598293727 p^{2} T^{7} + 51119807 p^{4} T^{8} + 181142 p^{6} T^{9} - 7367 p^{8} T^{10} - 55 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
61 \( 1 + 9201 T^{2} + 56941411 T^{4} + 154999649300 T^{6} - 75695642240335 T^{8} - 3911957905117164149 T^{10} - \)\(19\!\cdots\!14\)\( T^{12} - 3911957905117164149 p^{4} T^{14} - 75695642240335 p^{8} T^{16} + 154999649300 p^{12} T^{18} + 56941411 p^{16} T^{20} + 9201 p^{20} T^{22} + p^{24} T^{24} \)
67 \( ( 1 + 217 T + 18053 T^{2} + 1666970 T^{3} + 209974835 T^{4} + 15078221277 T^{5} + 815515698066 T^{6} + 15078221277 p^{2} T^{7} + 209974835 p^{4} T^{8} + 1666970 p^{6} T^{9} + 18053 p^{8} T^{10} + 217 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
71 \( ( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 245626031 p^{4} T^{8} - 23062 p^{8} T^{10} + p^{12} T^{12} )^{2} \)
73 \( ( 1 - 51 T - 9165 T^{2} + 579660 T^{3} + 45654729 T^{4} - 1993134921 T^{5} - 160997144218 T^{6} - 1993134921 p^{2} T^{7} + 45654729 p^{4} T^{8} + 579660 p^{6} T^{9} - 9165 p^{8} T^{10} - 51 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
79 \( 1 + 16693 T^{2} + 149251283 T^{4} + 640487711012 T^{6} - 978609215845699 T^{8} - 44625107831251066425 T^{10} - \)\(37\!\cdots\!74\)\( T^{12} - 44625107831251066425 p^{4} T^{14} - 978609215845699 p^{8} T^{16} + 640487711012 p^{12} T^{18} + 149251283 p^{16} T^{20} + 16693 p^{20} T^{22} + p^{24} T^{24} \)
83 \( ( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 22583 p^{2} T^{4} - 134 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
89 \( ( 1 - 107 T + 1395 T^{2} + 497084 T^{3} - 62702935 T^{4} + 2422278015 T^{5} + 93422604838 T^{6} + 2422278015 p^{2} T^{7} - 62702935 p^{4} T^{8} + 497084 p^{6} T^{9} + 1395 p^{8} T^{10} - 107 p^{10} T^{11} + p^{12} T^{12} )^{2} \)
97 \( ( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 25191 p^{2} T^{4} + 38 p^{4} T^{5} + p^{6} T^{6} )^{4} \)
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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

−4.00410611935418369707450856146, −3.89643372259910247924388276240, −3.61655452841785874109024417000, −3.57463403513901454153848840201, −3.50665329629585684070345654213, −3.45668677393810565548505701874, −3.34922010994147432579079615024, −3.10675666968619845541959213014, −3.06400413257715554281916241307, −2.65867871163712756028765098453, −2.65715253184372298640245696898, −2.61157855964840144902095345486, −2.32548370289706160110479165417, −2.27929703696035106858965262291, −2.16828732797048316427962863867, −2.12851134836776166060096886117, −2.09353513736377760644131254398, −1.73015094523711141262642636064, −1.67326049554080015315210636667, −1.46634765119382521410845708182, −1.36605083453434921383499543757, −0.74487119924691309227906736146, −0.66116450474047397058457746148, −0.39435241098187786492668510456, −0.11371656378329650363855742507, 0.11371656378329650363855742507, 0.39435241098187786492668510456, 0.66116450474047397058457746148, 0.74487119924691309227906736146, 1.36605083453434921383499543757, 1.46634765119382521410845708182, 1.67326049554080015315210636667, 1.73015094523711141262642636064, 2.09353513736377760644131254398, 2.12851134836776166060096886117, 2.16828732797048316427962863867, 2.27929703696035106858965262291, 2.32548370289706160110479165417, 2.61157855964840144902095345486, 2.65715253184372298640245696898, 2.65867871163712756028765098453, 3.06400413257715554281916241307, 3.10675666968619845541959213014, 3.34922010994147432579079615024, 3.45668677393810565548505701874, 3.50665329629585684070345654213, 3.57463403513901454153848840201, 3.61655452841785874109024417000, 3.89643372259910247924388276240, 4.00410611935418369707450856146

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

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