Properties

Label 10-6080e5-1.1-c1e5-0-1
Degree $10$
Conductor $8.308\times 10^{18}$
Sign $1$
Analytic cond. $2.69713\times 10^{8}$
Root an. cond. $6.96771$
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  + 4·3-s + 5·5-s + 4·7-s + 4·9-s + 2·11-s + 4·13-s + 20·15-s − 12·17-s − 5·19-s + 16·21-s + 8·23-s + 15·25-s − 4·27-s + 6·29-s + 10·31-s + 8·33-s + 20·35-s + 6·37-s + 16·39-s − 8·41-s + 12·43-s + 20·45-s + 16·47-s − 6·49-s − 48·51-s + 18·53-s + 10·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 2.23·5-s + 1.51·7-s + 4/3·9-s + 0.603·11-s + 1.10·13-s + 5.16·15-s − 2.91·17-s − 1.14·19-s + 3.49·21-s + 1.66·23-s + 3·25-s − 0.769·27-s + 1.11·29-s + 1.79·31-s + 1.39·33-s + 3.38·35-s + 0.986·37-s + 2.56·39-s − 1.24·41-s + 1.82·43-s + 2.98·45-s + 2.33·47-s − 6/7·49-s − 6.72·51-s + 2.47·53-s + 1.34·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{5} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{30} \cdot 5^{5} \cdot 19^{5}\)
Sign: $1$
Analytic conductor: \(2.69713\times 10^{8}\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{30} \cdot 5^{5} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(84.18723343\)
\(L(\frac12)\) \(\approx\) \(84.18723343\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{5} \)
19$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - 4 T + 4 p T^{2} - 28 T^{3} + 55 T^{4} - 106 T^{5} + 55 p T^{6} - 28 p^{2} T^{7} + 4 p^{4} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 4 T + 22 T^{2} - 52 T^{3} + 181 T^{4} - 340 T^{5} + 181 p T^{6} - 52 p^{2} T^{7} + 22 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T + 23 T^{2} - 4 p T^{3} + 362 T^{4} - 420 T^{5} + 362 p T^{6} - 4 p^{3} T^{7} + 23 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 46 T^{2} - 134 T^{3} + 895 T^{4} - 2134 T^{5} + 895 p T^{6} - 134 p^{2} T^{7} + 46 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 12 T + 98 T^{2} + 514 T^{3} + 2349 T^{4} + 9348 T^{5} + 2349 p T^{6} + 514 p^{2} T^{7} + 98 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 8 T + 78 T^{2} - 584 T^{3} + 3493 T^{4} - 17628 T^{5} + 3493 p T^{6} - 584 p^{2} T^{7} + 78 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 6 T + 64 T^{2} - 106 T^{3} + 1655 T^{4} - 1536 T^{5} + 1655 p T^{6} - 106 p^{2} T^{7} + 64 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 10 T + 115 T^{2} - 760 T^{3} + 5458 T^{4} - 29788 T^{5} + 5458 p T^{6} - 760 p^{2} T^{7} + 115 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 6 T + 87 T^{2} - 526 T^{3} + 5556 T^{4} - 24712 T^{5} + 5556 p T^{6} - 526 p^{2} T^{7} + 87 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 8 T + 193 T^{2} + 1152 T^{3} + 15206 T^{4} + 67888 T^{5} + 15206 p T^{6} + 1152 p^{2} T^{7} + 193 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 12 T + 107 T^{2} - 1148 T^{3} + 8942 T^{4} - 53024 T^{5} + 8942 p T^{6} - 1148 p^{2} T^{7} + 107 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 16 T + 255 T^{2} - 2204 T^{3} + 20430 T^{4} - 129896 T^{5} + 20430 p T^{6} - 2204 p^{2} T^{7} + 255 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 18 T + 186 T^{2} - 1916 T^{3} + 17247 T^{4} - 130458 T^{5} + 17247 p T^{6} - 1916 p^{2} T^{7} + 186 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 8 T + 240 T^{2} - 1750 T^{3} + 25327 T^{4} - 151196 T^{5} + 25327 p T^{6} - 1750 p^{2} T^{7} + 240 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 2 T + 277 T^{2} + 508 T^{3} + 32150 T^{4} + 47028 T^{5} + 32150 p T^{6} + 508 p^{2} T^{7} + 277 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 10 T + 256 T^{2} - 2166 T^{3} + 28507 T^{4} - 200414 T^{5} + 28507 p T^{6} - 2166 p^{2} T^{7} + 256 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 18 T + 159 T^{2} + 1024 T^{3} + 2774 T^{4} - 14052 T^{5} + 2774 p T^{6} + 1024 p^{2} T^{7} + 159 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 28 T + 618 T^{2} + 8822 T^{3} + 107301 T^{4} + 984108 T^{5} + 107301 p T^{6} + 8822 p^{2} T^{7} + 618 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 14 T + 355 T^{2} - 3416 T^{3} + 50770 T^{4} - 368276 T^{5} + 50770 p T^{6} - 3416 p^{2} T^{7} + 355 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 8 T + 211 T^{2} - 2484 T^{3} + 23022 T^{4} - 296280 T^{5} + 23022 p T^{6} - 2484 p^{2} T^{7} + 211 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 30 T + 677 T^{2} + 10744 T^{3} + 140338 T^{4} + 1436436 T^{5} + 140338 p T^{6} + 10744 p^{2} T^{7} + 677 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 18 T + 359 T^{2} + 4250 T^{3} + 60944 T^{4} + 586328 T^{5} + 60944 p T^{6} + 4250 p^{2} T^{7} + 359 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.61290459196152708463568108107, −4.43651776910465323165503638201, −4.40548933699646528307352793461, −4.31960381067805832117646393261, −4.31224204510337163619006014664, −4.07020662580350112082193078542, −3.61482390338805504454729189795, −3.60283436426330752055759599949, −3.46570836443418083716668959372, −3.24203962183610341734728730455, −2.87611055352162256480396468265, −2.78710467386997000120806640909, −2.63712544612822535941808946890, −2.61551997333010992654600081429, −2.59802015547729433316790380890, −2.32279983149029754413364944079, −2.03512782353930172438065076411, −1.94239544962094570968932147477, −1.73686685906974836148517358823, −1.62682249471496282441888944854, −1.39500721012441549276771296797, −1.15482233636226196929798596924, −0.71148670818245232967518408785, −0.66516431469268541983840734608, −0.54791363524363720304891799546, 0.54791363524363720304891799546, 0.66516431469268541983840734608, 0.71148670818245232967518408785, 1.15482233636226196929798596924, 1.39500721012441549276771296797, 1.62682249471496282441888944854, 1.73686685906974836148517358823, 1.94239544962094570968932147477, 2.03512782353930172438065076411, 2.32279983149029754413364944079, 2.59802015547729433316790380890, 2.61551997333010992654600081429, 2.63712544612822535941808946890, 2.78710467386997000120806640909, 2.87611055352162256480396468265, 3.24203962183610341734728730455, 3.46570836443418083716668959372, 3.60283436426330752055759599949, 3.61482390338805504454729189795, 4.07020662580350112082193078542, 4.31224204510337163619006014664, 4.31960381067805832117646393261, 4.40548933699646528307352793461, 4.43651776910465323165503638201, 4.61290459196152708463568108107

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