Properties

Degree 40
Conductor $ 2^{60} \cdot 3^{60} \cdot 7^{20} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 20·7-s − 3·16-s + 36·25-s + 20·28-s + 36·31-s + 210·49-s + 7·64-s + 64·79-s + 56·97-s − 36·100-s − 12·103-s + 60·112-s + 68·121-s − 36·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 166·169-s + 173-s − 720·175-s + ⋯
L(s)  = 1  − 1/2·4-s − 7.55·7-s − 3/4·16-s + 36/5·25-s + 3.77·28-s + 6.46·31-s + 30·49-s + 7/8·64-s + 7.20·79-s + 5.68·97-s − 3.59·100-s − 1.18·103-s + 5.66·112-s + 6.18·121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 12.7·169-s + 0.0760·173-s − 54.4·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{60} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 3^{60} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(40\)
\( N \)  =  \(2^{60} \cdot 3^{60} \cdot 7^{20}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1512} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(40,\ 2^{60} \cdot 3^{60} \cdot 7^{20} ,\ ( \ : [1/2]^{20} ),\ 1 )$
$L(1)$  $\approx$  $1.103934326$
$L(\frac12)$  $\approx$  $1.103934326$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 40. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 39.
$p$$F_p(T)$
bad2 \( 1 + T^{2} + p^{2} T^{4} + p^{3} T^{8} + p^{2} T^{10} + p^{5} T^{12} + p^{8} T^{16} + p^{8} T^{18} + p^{10} T^{20} \)
3 \( 1 \)
7 \( ( 1 + T )^{20} \)
good5 \( ( 1 - 18 T^{2} + 187 T^{4} - 1608 T^{6} + 10781 T^{8} - 57906 T^{10} + 10781 p^{2} T^{12} - 1608 p^{4} T^{14} + 187 p^{6} T^{16} - 18 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
11 \( ( 1 - 34 T^{2} + 589 T^{4} - 8472 T^{6} + 122234 T^{8} - 1528172 T^{10} + 122234 p^{2} T^{12} - 8472 p^{4} T^{14} + 589 p^{6} T^{16} - 34 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
13 \( ( 1 - 83 T^{2} + 3533 T^{4} - 98556 T^{6} + 1975738 T^{8} - 29558418 T^{10} + 1975738 p^{2} T^{12} - 98556 p^{4} T^{14} + 3533 p^{6} T^{16} - 83 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
17 \( ( 1 + 39 T^{2} + 455 T^{4} - 2382 T^{6} - 47987 T^{8} + 145305 T^{10} - 47987 p^{2} T^{12} - 2382 p^{4} T^{14} + 455 p^{6} T^{16} + 39 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
19 \( ( 1 - 70 T^{2} + 175 p T^{4} - 107448 T^{6} + 2823098 T^{8} - 58351460 T^{10} + 2823098 p^{2} T^{12} - 107448 p^{4} T^{14} + 175 p^{7} T^{16} - 70 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
23 \( ( 1 + 139 T^{2} + 10081 T^{4} + 21108 p T^{6} + 16965806 T^{8} + 447075730 T^{10} + 16965806 p^{2} T^{12} + 21108 p^{5} T^{14} + 10081 p^{6} T^{16} + 139 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
29 \( ( 1 - 189 T^{2} + 17145 T^{4} - 999612 T^{6} + 42327774 T^{8} - 1383568894 T^{10} + 42327774 p^{2} T^{12} - 999612 p^{4} T^{14} + 17145 p^{6} T^{16} - 189 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
31 \( ( 1 - 9 T + 171 T^{2} - 1092 T^{3} + 11058 T^{4} - 50402 T^{5} + 11058 p T^{6} - 1092 p^{2} T^{7} + 171 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
37 \( ( 1 - 22 T^{2} + 1771 T^{4} + 13920 T^{6} + 526445 T^{8} + 112600450 T^{10} + 526445 p^{2} T^{12} + 13920 p^{4} T^{14} + 1771 p^{6} T^{16} - 22 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
41 \( ( 1 + 190 T^{2} + 16651 T^{4} + 899952 T^{6} + 36306749 T^{8} + 1400716342 T^{10} + 36306749 p^{2} T^{12} + 899952 p^{4} T^{14} + 16651 p^{6} T^{16} + 190 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
43 \( ( 1 - 131 T^{2} + 5615 T^{4} - 93162 T^{6} + 5954173 T^{8} - 455929005 T^{10} + 5954173 p^{2} T^{12} - 93162 p^{4} T^{14} + 5615 p^{6} T^{16} - 131 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
47 \( ( 1 + 110 T^{2} + 11187 T^{4} + 742128 T^{6} + 47768349 T^{8} + 2326511142 T^{10} + 47768349 p^{2} T^{12} + 742128 p^{4} T^{14} + 11187 p^{6} T^{16} + 110 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
53 \( ( 1 - 193 T^{2} + 20557 T^{4} - 1818204 T^{6} + 128583722 T^{8} - 7326778022 T^{10} + 128583722 p^{2} T^{12} - 1818204 p^{4} T^{14} + 20557 p^{6} T^{16} - 193 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
59 \( ( 1 - 449 T^{2} + 96439 T^{4} - 13053078 T^{6} + 1232082461 T^{8} - 84671750375 T^{10} + 1232082461 p^{2} T^{12} - 13053078 p^{4} T^{14} + 96439 p^{6} T^{16} - 449 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
61 \( ( 1 - 186 T^{2} + 23517 T^{4} - 2152152 T^{6} + 175128378 T^{8} - 188535020 p T^{10} + 175128378 p^{2} T^{12} - 2152152 p^{4} T^{14} + 23517 p^{6} T^{16} - 186 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
67 \( ( 1 - 495 T^{2} + 120093 T^{4} - 18487716 T^{6} + 1986970986 T^{8} - 155229602234 T^{10} + 1986970986 p^{2} T^{12} - 18487716 p^{4} T^{14} + 120093 p^{6} T^{16} - 495 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
71 \( ( 1 + 343 T^{2} + 52613 T^{4} + 4802116 T^{6} + 309151594 T^{8} + 19321038474 T^{10} + 309151594 p^{2} T^{12} + 4802116 p^{4} T^{14} + 52613 p^{6} T^{16} + 343 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
73 \( ( 1 + 149 T^{2} + 432 T^{3} + 14410 T^{4} + 51408 T^{5} + 14410 p T^{6} + 432 p^{2} T^{7} + 149 p^{3} T^{8} + p^{5} T^{10} )^{4} \)
79 \( ( 1 - 16 T + 335 T^{2} - 3202 T^{3} + 40117 T^{4} - 296322 T^{5} + 40117 p T^{6} - 3202 p^{2} T^{7} + 335 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
83 \( ( 1 - 138 T^{2} + 13627 T^{4} - 1173792 T^{6} + 994015 p T^{8} - 3517947618 T^{10} + 994015 p^{3} T^{12} - 1173792 p^{4} T^{14} + 13627 p^{6} T^{16} - 138 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
89 \( ( 1 + 547 T^{2} + 153205 T^{4} + 28413396 T^{6} + 3828024122 T^{8} + 389779473874 T^{10} + 3828024122 p^{2} T^{12} + 28413396 p^{4} T^{14} + 153205 p^{6} T^{16} + 547 p^{8} T^{18} + p^{10} T^{20} )^{2} \)
97 \( ( 1 - 14 T + 341 T^{2} - 3480 T^{3} + 53038 T^{4} - 434964 T^{5} + 53038 p T^{6} - 3480 p^{2} T^{7} + 341 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.09926923845619842078884369969, −2.05060292219297249234001177414, −2.04874003425959753983262074330, −1.78574651509908858551018899872, −1.76524561892432315484268807150, −1.73342674523395717640201322531, −1.66527094996962878284461031365, −1.63164554247219376232966888506, −1.47036799834461974390045189593, −1.25229572277432428014701730170, −1.22405786708977729674115013901, −1.07183800269576097156063959561, −1.06815861232807092487021813488, −1.03723454490197360060397971843, −1.03309800476160501529880668075, −0.954412163671556532903735941540, −0.821706155197072658631174586551, −0.74454432723300926003519802207, −0.62493901999726350076248227045, −0.58652345978502163813270480630, −0.51202676466686386047631290982, −0.47180545967596727826058612331, −0.44763152747786232055372405587, −0.32148682811248410987484957936, −0.03656907384532636502011256172, 0.03656907384532636502011256172, 0.32148682811248410987484957936, 0.44763152747786232055372405587, 0.47180545967596727826058612331, 0.51202676466686386047631290982, 0.58652345978502163813270480630, 0.62493901999726350076248227045, 0.74454432723300926003519802207, 0.821706155197072658631174586551, 0.954412163671556532903735941540, 1.03309800476160501529880668075, 1.03723454490197360060397971843, 1.06815861232807092487021813488, 1.07183800269576097156063959561, 1.22405786708977729674115013901, 1.25229572277432428014701730170, 1.47036799834461974390045189593, 1.63164554247219376232966888506, 1.66527094996962878284461031365, 1.73342674523395717640201322531, 1.76524561892432315484268807150, 1.78574651509908858551018899872, 2.04874003425959753983262074330, 2.05060292219297249234001177414, 2.09926923845619842078884369969

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

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