Properties

Label 10-4008e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.034\times 10^{18}$
Sign $-1$
Analytic cond. $3.35756\times 10^{7}$
Root an. cond. $5.65721$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

Downloads

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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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{5} \cdot 167^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 3^{5} \cdot 167^{5}\)
Sign: $-1$
Analytic conductor: \(3.35756\times 10^{7}\)
Root analytic conductor: \(5.65721\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{15} \cdot 3^{5} \cdot 167^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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

−5.30022856170369915555394197423, −5.09961074424741458684954432017, −5.02043240214838772474515458337, −4.77179535365337229207107596371, −4.63300767407705285138933564820, −4.44359276806735478585300794704, −4.36942156170902509385263612701, −4.32100445522147252707512103048, −4.27721790590279422552743450436, −4.06606005984240688948464176897, −3.50233815210036930219859530438, −3.33946501928474368776207139599, −3.25655154483527605395352090965, −3.19411980797083537811340659605, −3.07390069788395908919158730894, −2.80863085782978825772713865253, −2.67645203666556476856313853911, −2.56232694150682437464519344108, −2.34152839061225936286118665823, −2.28790195023889265508485217116, −1.87083100646288175454834272209, −1.72149717070072404008678544923, −1.67927871121022525576417319446, −1.48986388214404931652563979471, −1.36328730862717572190691733952, 0, 0, 0, 0, 0, 1.36328730862717572190691733952, 1.48986388214404931652563979471, 1.67927871121022525576417319446, 1.72149717070072404008678544923, 1.87083100646288175454834272209, 2.28790195023889265508485217116, 2.34152839061225936286118665823, 2.56232694150682437464519344108, 2.67645203666556476856313853911, 2.80863085782978825772713865253, 3.07390069788395908919158730894, 3.19411980797083537811340659605, 3.25655154483527605395352090965, 3.33946501928474368776207139599, 3.50233815210036930219859530438, 4.06606005984240688948464176897, 4.27721790590279422552743450436, 4.32100445522147252707512103048, 4.36942156170902509385263612701, 4.44359276806735478585300794704, 4.63300767407705285138933564820, 4.77179535365337229207107596371, 5.02043240214838772474515458337, 5.09961074424741458684954432017, 5.30022856170369915555394197423

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