Properties

Label 10-2960e5-1.1-c1e5-0-2
Degree $10$
Conductor $2.272\times 10^{17}$
Sign $1$
Analytic cond. $7.37639\times 10^{6}$
Root an. cond. $4.86165$
Motivic weight $1$
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  + 5·3-s + 5·5-s + 5·7-s + 8·9-s + 7·11-s − 6·13-s + 25·15-s + 12·19-s + 25·21-s + 6·23-s + 15·25-s − 27-s + 6·29-s + 10·31-s + 35·33-s + 25·35-s − 5·37-s − 30·39-s + 7·41-s + 22·43-s + 40·45-s + 13·47-s + 2·49-s − 11·53-s + 35·55-s + 60·57-s + 10·59-s + ⋯
L(s)  = 1  + 2.88·3-s + 2.23·5-s + 1.88·7-s + 8/3·9-s + 2.11·11-s − 1.66·13-s + 6.45·15-s + 2.75·19-s + 5.45·21-s + 1.25·23-s + 3·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 6.09·33-s + 4.22·35-s − 0.821·37-s − 4.80·39-s + 1.09·41-s + 3.35·43-s + 5.96·45-s + 1.89·47-s + 2/7·49-s − 1.51·53-s + 4.71·55-s + 7.94·57-s + 1.30·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 5^{5} \cdot 37^{5}\)
Sign: $1$
Analytic conductor: \(7.37639\times 10^{6}\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{20} \cdot 5^{5} \cdot 37^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(114.3272780\)
\(L(\frac12)\) \(\approx\) \(114.3272780\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{5} \)
37$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - 5 T + 17 T^{2} - 44 T^{3} + 100 T^{4} - 188 T^{5} + 100 p T^{6} - 44 p^{2} T^{7} + 17 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 5 T + 23 T^{2} - 40 T^{3} + 78 T^{4} - 8 T^{5} + 78 p T^{6} - 40 p^{2} T^{7} + 23 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 7 T + 5 p T^{2} - 260 T^{3} + 1226 T^{4} - 4058 T^{5} + 1226 p T^{6} - 260 p^{2} T^{7} + 5 p^{4} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 6 T + 37 T^{2} + 140 T^{3} + 402 T^{4} + 1604 T^{5} + 402 p T^{6} + 140 p^{2} T^{7} + 37 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 9 T^{2} - 100 T^{3} + 282 T^{4} - 504 T^{5} + 282 p T^{6} - 100 p^{2} T^{7} + 9 p^{3} T^{8} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 12 T + 121 T^{2} - 850 T^{3} + 5046 T^{4} - 23668 T^{5} + 5046 p T^{6} - 850 p^{2} T^{7} + 121 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 6 T + 67 T^{2} - 120 T^{3} + 1098 T^{4} + 1052 T^{5} + 1098 p T^{6} - 120 p^{2} T^{7} + 67 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 6 T + 97 T^{2} - 264 T^{3} + 3354 T^{4} - 4996 T^{5} + 3354 p T^{6} - 264 p^{2} T^{7} + 97 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 10 T + 145 T^{2} - 870 T^{3} + 7474 T^{4} - 33608 T^{5} + 7474 p T^{6} - 870 p^{2} T^{7} + 145 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 7 T + 5 p T^{2} - 1100 T^{3} + 16826 T^{4} - 66698 T^{5} + 16826 p T^{6} - 1100 p^{2} T^{7} + 5 p^{4} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 22 T + 367 T^{2} - 4168 T^{3} + 38338 T^{4} - 277060 T^{5} + 38338 p T^{6} - 4168 p^{2} T^{7} + 367 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 13 T + 259 T^{2} - 2168 T^{3} + 24246 T^{4} - 145120 T^{5} + 24246 p T^{6} - 2168 p^{2} T^{7} + 259 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 11 T + 73 T^{2} - 188 T^{3} - 30 p T^{4} - 23182 T^{5} - 30 p^{2} T^{6} - 188 p^{2} T^{7} + 73 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 10 T + 253 T^{2} - 1874 T^{3} + 27242 T^{4} - 152960 T^{5} + 27242 p T^{6} - 1874 p^{2} T^{7} + 253 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 177 T^{2} + 144 T^{3} + 17738 T^{4} + 6752 T^{5} + 17738 p T^{6} + 144 p^{2} T^{7} + 177 p^{3} T^{8} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 24 T + 385 T^{2} - 4850 T^{3} + 50386 T^{4} - 435420 T^{5} + 50386 p T^{6} - 4850 p^{2} T^{7} + 385 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 13 T + 143 T^{2} - 1068 T^{3} + 14598 T^{4} - 119726 T^{5} + 14598 p T^{6} - 1068 p^{2} T^{7} + 143 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 13 T + 257 T^{2} + 2884 T^{3} + 32406 T^{4} + 294142 T^{5} + 32406 p T^{6} + 2884 p^{2} T^{7} + 257 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 2 T + 285 T^{2} + 286 T^{3} + 39270 T^{4} + 31680 T^{5} + 39270 p T^{6} + 286 p^{2} T^{7} + 285 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 13 T + 161 T^{2} - 2704 T^{3} + 28628 T^{4} - 234316 T^{5} + 28628 p T^{6} - 2704 p^{2} T^{7} + 161 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 8 T + 301 T^{2} - 3120 T^{3} + 41770 T^{4} - 426640 T^{5} + 41770 p T^{6} - 3120 p^{2} T^{7} + 301 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 20 T + 405 T^{2} + 4224 T^{3} + 51002 T^{4} + 411352 T^{5} + 51002 p T^{6} + 4224 p^{2} T^{7} + 405 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.09498721469517367504986536248, −5.06362876927922622370074677802, −4.76859581132167961966207955843, −4.76268660268179950266234507624, −4.66911445639974772699884828000, −4.26236822829764985753743501001, −4.15288613498971970057019186059, −3.97982514499059464350540393590, −3.77686783366891145878287078657, −3.65036214397222232239846010032, −3.15117425739817709490130486258, −3.11981205762919448944044484362, −2.96671525834612210855475185995, −2.92579910871870778824239966595, −2.81182633985447652742022627245, −2.33804573798431334728166562072, −2.32269191579680501058468020894, −2.21041187553481572909436610966, −2.07389743872694472129760557503, −1.82673669083948028323384468685, −1.36700253444496342444058046329, −1.28843432005595200464234226723, −0.931276742715972215131005861603, −0.923568043368975551631184026104, −0.74346679533909908159908447173, 0.74346679533909908159908447173, 0.923568043368975551631184026104, 0.931276742715972215131005861603, 1.28843432005595200464234226723, 1.36700253444496342444058046329, 1.82673669083948028323384468685, 2.07389743872694472129760557503, 2.21041187553481572909436610966, 2.32269191579680501058468020894, 2.33804573798431334728166562072, 2.81182633985447652742022627245, 2.92579910871870778824239966595, 2.96671525834612210855475185995, 3.11981205762919448944044484362, 3.15117425739817709490130486258, 3.65036214397222232239846010032, 3.77686783366891145878287078657, 3.97982514499059464350540393590, 4.15288613498971970057019186059, 4.26236822829764985753743501001, 4.66911445639974772699884828000, 4.76268660268179950266234507624, 4.76859581132167961966207955843, 5.06362876927922622370074677802, 5.09498721469517367504986536248

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