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Properties

Label 10-5580e5-1.1-c1e5-0-0
Degree $10$
Conductor $5.410\times 10^{18}$
Sign $1$
Analytic cond. $1.75612\times 10^{8}$
Root an. cond. $6.67506$
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  − 5·5-s − 4·7-s + 2·11-s − 2·13-s + 6·17-s + 8·23-s + 15·25-s + 2·29-s − 5·31-s + 20·35-s − 14·37-s − 10·43-s + 12·47-s − 11·49-s + 4·53-s − 10·55-s + 10·59-s − 2·61-s + 10·65-s − 16·67-s + 12·71-s + 4·73-s − 8·77-s + 28·83-s − 30·85-s + 18·89-s + 8·91-s + ⋯
L(s)  = 1  − 2.23·5-s − 1.51·7-s + 0.603·11-s − 0.554·13-s + 1.45·17-s + 1.66·23-s + 3·25-s + 0.371·29-s − 0.898·31-s + 3.38·35-s − 2.30·37-s − 1.52·43-s + 1.75·47-s − 1.57·49-s + 0.549·53-s − 1.34·55-s + 1.30·59-s − 0.256·61-s + 1.24·65-s − 1.95·67-s + 1.42·71-s + 0.468·73-s − 0.911·77-s + 3.07·83-s − 3.25·85-s + 1.90·89-s + 0.838·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 5^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{10} \cdot 5^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 3^{10} \cdot 5^{5} \cdot 31^{5}\)
Sign: $1$
Analytic conductor: \(1.75612\times 10^{8}\)
Root analytic conductor: \(6.67506\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{10} \cdot 3^{10} \cdot 5^{5} \cdot 31^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.042605826\)
\(L(\frac12)\) \(\approx\) \(4.042605826\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{5} \)
31$C_1$ \( ( 1 + T )^{5} \)
good7$C_2 \wr S_5$ \( 1 + 4 T + 27 T^{2} + 76 T^{3} + 328 T^{4} + 102 p T^{5} + 328 p T^{6} + 76 p^{2} T^{7} + 27 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T + 23 T^{2} - 72 T^{3} + 410 T^{4} - 868 T^{5} + 410 p T^{6} - 72 p^{2} T^{7} + 23 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 2 T + 21 T^{2} + 86 T^{3} + 472 T^{4} + 1098 T^{5} + 472 p T^{6} + 86 p^{2} T^{7} + 21 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 6 T + 31 T^{2} - 20 T^{3} - 548 T^{4} + 3136 T^{5} - 548 p T^{6} - 20 p^{2} T^{7} + 31 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 35 T^{2} + 136 T^{3} + 10 p T^{4} + 5120 T^{5} + 10 p^{2} T^{6} + 136 p^{2} T^{7} + 35 p^{3} T^{8} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 8 T + 65 T^{2} - 180 T^{3} + 488 T^{4} + 1028 T^{5} + 488 p T^{6} - 180 p^{2} T^{7} + 65 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 2 T + 119 T^{2} - 234 T^{3} + 6248 T^{4} - 10186 T^{5} + 6248 p T^{6} - 234 p^{2} T^{7} + 119 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 14 T + 189 T^{2} + 1732 T^{3} + 14384 T^{4} + 90630 T^{5} + 14384 p T^{6} + 1732 p^{2} T^{7} + 189 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 169 T^{2} + 48 T^{3} + 12526 T^{4} + 3720 T^{5} + 12526 p T^{6} + 48 p^{2} T^{7} + 169 p^{3} T^{8} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 10 T + 171 T^{2} + 1340 T^{3} + 13406 T^{4} + 80748 T^{5} + 13406 p T^{6} + 1340 p^{2} T^{7} + 171 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 12 T + 217 T^{2} - 1764 T^{3} + 18652 T^{4} - 113556 T^{5} + 18652 p T^{6} - 1764 p^{2} T^{7} + 217 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 4 T + 167 T^{2} - 832 T^{3} + 14388 T^{4} - 62356 T^{5} + 14388 p T^{6} - 832 p^{2} T^{7} + 167 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 10 T + 305 T^{2} - 2230 T^{3} + 36324 T^{4} - 193678 T^{5} + 36324 p T^{6} - 2230 p^{2} T^{7} + 305 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 2 T + 153 T^{2} - 376 T^{3} + 10450 T^{4} - 51012 T^{5} + 10450 p T^{6} - 376 p^{2} T^{7} + 153 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 16 T + 375 T^{2} + 3990 T^{3} + 52176 T^{4} + 392790 T^{5} + 52176 p T^{6} + 3990 p^{2} T^{7} + 375 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 12 T + 217 T^{2} - 1980 T^{3} + 22708 T^{4} - 183630 T^{5} + 22708 p T^{6} - 1980 p^{2} T^{7} + 217 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 4 T + 225 T^{2} - 952 T^{3} + 25492 T^{4} - 101202 T^{5} + 25492 p T^{6} - 952 p^{2} T^{7} + 225 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 191 T^{2} + 388 T^{3} + 20542 T^{4} + 41636 T^{5} + 20542 p T^{6} + 388 p^{2} T^{7} + 191 p^{3} T^{8} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 28 T + 497 T^{2} - 6992 T^{3} + 80620 T^{4} - 774660 T^{5} + 80620 p T^{6} - 6992 p^{2} T^{7} + 497 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 18 T + 379 T^{2} - 3900 T^{3} + 50608 T^{4} - 399990 T^{5} + 50608 p T^{6} - 3900 p^{2} T^{7} + 379 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 16 T + 417 T^{2} + 5676 T^{3} + 76878 T^{4} + 797280 T^{5} + 76878 p T^{6} + 5676 p^{2} T^{7} + 417 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.72966538298013035395702822033, −4.48620540764232924804363664805, −4.45979249131373168622163165011, −4.44104228509179487191408091568, −4.17861527845647730885650515454, −3.83812355684058480405614524555, −3.61732857419078919495603470099, −3.61539697654699774477659139453, −3.59999842549136693954934912187, −3.55034036778496196511118092936, −3.08627645847920651369217570613, −2.94077938055572179797837891365, −2.92113582895318632384356191258, −2.90640006108037055322578771281, −2.84675508607187981116578810556, −2.07382098976685792544777642712, −1.93077469888653639670210564744, −1.91524777277372849811129701101, −1.79180334000261434157126924857, −1.55451115470997304906668909321, −0.992050642123211909129640772682, −0.793964318714229831748264001284, −0.57159717076268274934508304635, −0.55678242796015760243336301855, −0.36298336557408518955538362549, 0.36298336557408518955538362549, 0.55678242796015760243336301855, 0.57159717076268274934508304635, 0.793964318714229831748264001284, 0.992050642123211909129640772682, 1.55451115470997304906668909321, 1.79180334000261434157126924857, 1.91524777277372849811129701101, 1.93077469888653639670210564744, 2.07382098976685792544777642712, 2.84675508607187981116578810556, 2.90640006108037055322578771281, 2.92113582895318632384356191258, 2.94077938055572179797837891365, 3.08627645847920651369217570613, 3.55034036778496196511118092936, 3.59999842549136693954934912187, 3.61539697654699774477659139453, 3.61732857419078919495603470099, 3.83812355684058480405614524555, 4.17861527845647730885650515454, 4.44104228509179487191408091568, 4.45979249131373168622163165011, 4.48620540764232924804363664805, 4.72966538298013035395702822033

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