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Properties

Label 10-8016e5-1.1-c1e5-0-4
Degree $10$
Conductor $3.310\times 10^{19}$
Sign $-1$
Analytic cond. $1.07441\times 10^{9}$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s + 5-s + 4·7-s + 15·9-s + 3·11-s − 14·13-s − 5·15-s − 13·17-s + 2·19-s − 20·21-s + 5·23-s − 11·25-s − 35·27-s + 13·29-s − 2·31-s − 15·33-s + 4·35-s − 5·37-s + 70·39-s − 20·41-s + 20·43-s + 15·45-s − 47-s − 14·49-s + 65·51-s − 3·53-s + 3·55-s + ⋯
L(s)  = 1  − 2.88·3-s + 0.447·5-s + 1.51·7-s + 5·9-s + 0.904·11-s − 3.88·13-s − 1.29·15-s − 3.15·17-s + 0.458·19-s − 4.36·21-s + 1.04·23-s − 2.19·25-s − 6.73·27-s + 2.41·29-s − 0.359·31-s − 2.61·33-s + 0.676·35-s − 0.821·37-s + 11.2·39-s − 3.12·41-s + 3.04·43-s + 2.23·45-s − 0.145·47-s − 2·49-s + 9.10·51-s − 0.412·53-s + 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 167^{5}\)
Sign: $-1$
Analytic conductor: \(1.07441\times 10^{9}\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{5} \)
167$C_1$ \( ( 1 + T )^{5} \)
good5$C_2 \wr S_5$ \( 1 - T + 12 T^{2} - 8 T^{3} + 74 T^{4} - p^{2} T^{5} + 74 p T^{6} - 8 p^{2} T^{7} + 12 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 4 T + 30 T^{2} - 99 T^{3} + 402 T^{4} - 989 T^{5} + 402 p T^{6} - 99 p^{2} T^{7} + 30 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 3 T + 36 T^{2} - 102 T^{3} + 630 T^{4} - 13 p^{2} T^{5} + 630 p T^{6} - 102 p^{2} T^{7} + 36 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 14 T + 126 T^{2} + 790 T^{3} + 3937 T^{4} + 15593 T^{5} + 3937 p T^{6} + 790 p^{2} T^{7} + 126 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 13 T + 7 p T^{2} + 830 T^{3} + 4589 T^{4} + 20715 T^{5} + 4589 p T^{6} + 830 p^{2} T^{7} + 7 p^{4} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 2 T + 82 T^{2} - 108 T^{3} + 2829 T^{4} - 2651 T^{5} + 2829 p T^{6} - 108 p^{2} T^{7} + 82 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 5 T + 78 T^{2} - 386 T^{3} + 2904 T^{4} - 12749 T^{5} + 2904 p T^{6} - 386 p^{2} T^{7} + 78 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 13 T + 158 T^{2} - 1100 T^{3} + 7688 T^{4} - 39579 T^{5} + 7688 p T^{6} - 1100 p^{2} T^{7} + 158 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 2 T + 105 T^{2} + 215 T^{3} + 180 p T^{4} + 8789 T^{5} + 180 p^{2} T^{6} + 215 p^{2} T^{7} + 105 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 5 T + 114 T^{2} + 372 T^{3} + 5706 T^{4} + 14757 T^{5} + 5706 p T^{6} + 372 p^{2} T^{7} + 114 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 20 T + 219 T^{2} + 1955 T^{3} + 16226 T^{4} + 114713 T^{5} + 16226 p T^{6} + 1955 p^{2} T^{7} + 219 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 20 T + 293 T^{2} - 3203 T^{3} + 28482 T^{4} - 201705 T^{5} + 28482 p T^{6} - 3203 p^{2} T^{7} + 293 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + T + 80 T^{2} - 105 T^{3} + 5053 T^{4} + 627 T^{5} + 5053 p T^{6} - 105 p^{2} T^{7} + 80 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 3 T + 174 T^{2} + 423 T^{3} + 15171 T^{4} + 27597 T^{5} + 15171 p T^{6} + 423 p^{2} T^{7} + 174 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - T + 94 T^{2} + 389 T^{3} - 829 T^{4} + 52407 T^{5} - 829 p T^{6} + 389 p^{2} T^{7} + 94 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 34 T + 606 T^{2} + 7646 T^{3} + 77597 T^{4} + 659411 T^{5} + 77597 p T^{6} + 7646 p^{2} T^{7} + 606 p^{3} T^{8} + 34 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 16 T + 232 T^{2} - 2478 T^{3} + 23361 T^{4} - 190905 T^{5} + 23361 p T^{6} - 2478 p^{2} T^{7} + 232 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 5 T + 215 T^{2} - 595 T^{3} + 19727 T^{4} - 33999 T^{5} + 19727 p T^{6} - 595 p^{2} T^{7} + 215 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 12 T + 262 T^{2} + 1656 T^{3} + 24261 T^{4} + 111631 T^{5} + 24261 p T^{6} + 1656 p^{2} T^{7} + 262 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 20 T + 363 T^{2} - 4417 T^{3} + 51806 T^{4} - 475675 T^{5} + 51806 p T^{6} - 4417 p^{2} T^{7} + 363 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 15 T + 214 T^{2} - 1165 T^{3} + 7325 T^{4} + 47 p T^{5} + 7325 p T^{6} - 1165 p^{2} T^{7} + 214 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 48 T + 1306 T^{2} + 24216 T^{3} + 335755 T^{4} + 3585017 T^{5} + 335755 p T^{6} + 24216 p^{2} T^{7} + 1306 p^{3} T^{8} + 48 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 21 T + 593 T^{2} + 8159 T^{3} + 124997 T^{4} + 1187307 T^{5} + 124997 p T^{6} + 8159 p^{2} T^{7} + 593 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.89611815062004321645582322339, −4.80268555701265040425928261624, −4.79343868182720583719266007407, −4.61875440674409215501173522060, −4.61558580208166940505317508579, −4.29920035708106368788075390730, −4.18478385210971490968201846061, −4.02153062512842303586792354028, −3.98190898520338456372770195152, −3.90012637253628638187965128118, −3.37061730864962051933823796506, −3.23005870981125572225086411172, −3.07125174144606991826315152230, −2.85646135465915390618054476762, −2.75603209002922072620269804634, −2.46073064452010500204932704225, −2.24670329848124388112990588381, −2.09817366027946714201123983566, −2.01981264989404953358787082868, −1.97111754945221761394154249965, −1.56391358103938295904147870911, −1.31993303590551404742267345739, −1.31363043126904502381988635252, −1.03108880452256979196500769963, −0.937421823108086867613207509220, 0, 0, 0, 0, 0, 0.937421823108086867613207509220, 1.03108880452256979196500769963, 1.31363043126904502381988635252, 1.31993303590551404742267345739, 1.56391358103938295904147870911, 1.97111754945221761394154249965, 2.01981264989404953358787082868, 2.09817366027946714201123983566, 2.24670329848124388112990588381, 2.46073064452010500204932704225, 2.75603209002922072620269804634, 2.85646135465915390618054476762, 3.07125174144606991826315152230, 3.23005870981125572225086411172, 3.37061730864962051933823796506, 3.90012637253628638187965128118, 3.98190898520338456372770195152, 4.02153062512842303586792354028, 4.18478385210971490968201846061, 4.29920035708106368788075390730, 4.61558580208166940505317508579, 4.61875440674409215501173522060, 4.79343868182720583719266007407, 4.80268555701265040425928261624, 4.89611815062004321645582322339

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