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Properties

Label 10-2175e5-1.1-c1e5-0-2
Degree $10$
Conductor $4.867\times 10^{16}$
Sign $-1$
Analytic cond. $1.58009\times 10^{6}$
Root an. cond. $4.16742$
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  − 3·2-s − 5·3-s + 2·4-s + 15·6-s − 8·7-s + 2·8-s + 15·9-s + 12·11-s − 10·12-s − 2·13-s + 24·14-s − 4·16-s − 45·18-s − 2·19-s + 40·21-s − 36·22-s − 8·23-s − 10·24-s + 6·26-s − 35·27-s − 16·28-s + 5·29-s + 2·31-s + 8·32-s − 60·33-s + 30·36-s − 16·37-s + ⋯
L(s)  = 1  − 2.12·2-s − 2.88·3-s + 4-s + 6.12·6-s − 3.02·7-s + 0.707·8-s + 5·9-s + 3.61·11-s − 2.88·12-s − 0.554·13-s + 6.41·14-s − 16-s − 10.6·18-s − 0.458·19-s + 8.72·21-s − 7.67·22-s − 1.66·23-s − 2.04·24-s + 1.17·26-s − 6.73·27-s − 3.02·28-s + 0.928·29-s + 0.359·31-s + 1.41·32-s − 10.4·33-s + 5·36-s − 2.63·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 5^{10} \cdot 29^{5}\)
Sign: $-1$
Analytic conductor: \(1.58009\times 10^{6}\)
Root analytic conductor: \(4.16742\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{5} \cdot 5^{10} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3$C_1$ \( ( 1 + T )^{5} \)
5 \( 1 \)
29$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + 3 T + 7 T^{2} + 13 T^{3} + 23 T^{4} + 33 T^{5} + 23 p T^{6} + 13 p^{2} T^{7} + 7 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 8 T + 53 T^{2} + 230 T^{3} + 857 T^{4} + 2434 T^{5} + 857 p T^{6} + 230 p^{2} T^{7} + 53 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 12 T + 103 T^{2} - 600 T^{3} + 2829 T^{4} - 10300 T^{5} + 2829 p T^{6} - 600 p^{2} T^{7} + 103 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 2 T + 59 T^{2} + 94 T^{3} + 1461 T^{4} + 1772 T^{5} + 1461 p T^{6} + 94 p^{2} T^{7} + 59 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 57 T^{2} - 22 T^{3} + 1665 T^{4} - 450 T^{5} + 1665 p T^{6} - 22 p^{2} T^{7} + 57 p^{3} T^{8} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 2 T + 27 T^{2} + 132 T^{3} + 338 T^{4} + 172 p T^{5} + 338 p T^{6} + 132 p^{2} T^{7} + 27 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 8 T + 109 T^{2} + 676 T^{3} + 4912 T^{4} + 22672 T^{5} + 4912 p T^{6} + 676 p^{2} T^{7} + 109 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 2 T + 53 T^{2} + 16 T^{3} + 2248 T^{4} - 1468 T^{5} + 2248 p T^{6} + 16 p^{2} T^{7} + 53 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 16 T + 223 T^{2} + 2140 T^{3} + 17712 T^{4} + 115136 T^{5} + 17712 p T^{6} + 2140 p^{2} T^{7} + 223 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 14 T + 169 T^{2} + 1248 T^{3} + 10126 T^{4} + 61444 T^{5} + 10126 p T^{6} + 1248 p^{2} T^{7} + 169 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 69 T^{2} - 156 T^{3} + 4508 T^{4} - 6568 T^{5} + 4508 p T^{6} - 156 p^{2} T^{7} + 69 p^{3} T^{8} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T + 127 T^{2} + 342 T^{3} + 8633 T^{4} + 20620 T^{5} + 8633 p T^{6} + 342 p^{2} T^{7} + 127 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 26 T + 441 T^{2} + 5164 T^{3} + 49518 T^{4} + 386260 T^{5} + 49518 p T^{6} + 5164 p^{2} T^{7} + 441 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 4 T + 85 T^{2} - 404 T^{3} + 3924 T^{4} - 17824 T^{5} + 3924 p T^{6} - 404 p^{2} T^{7} + 85 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 12 T + 253 T^{2} + 2164 T^{3} + 27402 T^{4} + 181576 T^{5} + 27402 p T^{6} + 2164 p^{2} T^{7} + 253 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 12 T + 293 T^{2} + 2310 T^{3} + 33777 T^{4} + 200494 T^{5} + 33777 p T^{6} + 2310 p^{2} T^{7} + 293 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 30 T + 567 T^{2} - 7636 T^{3} + 84954 T^{4} - 775260 T^{5} + 84954 p T^{6} - 7636 p^{2} T^{7} + 567 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 12 T + 227 T^{2} - 2580 T^{3} + 25224 T^{4} - 252152 T^{5} + 25224 p T^{6} - 2580 p^{2} T^{7} + 227 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 18 T + 327 T^{2} + 44 p T^{3} + 44194 T^{4} + 376580 T^{5} + 44194 p T^{6} + 44 p^{3} T^{7} + 327 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 2 T + 309 T^{2} + 112 T^{3} + 41844 T^{4} - 8972 T^{5} + 41844 p T^{6} + 112 p^{2} T^{7} + 309 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 22 T + 421 T^{2} + 5760 T^{3} + 72333 T^{4} + 717438 T^{5} + 72333 p T^{6} + 5760 p^{2} T^{7} + 421 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 20 T + 629 T^{2} + 8228 T^{3} + 136670 T^{4} + 1220200 T^{5} + 136670 p T^{6} + 8228 p^{2} T^{7} + 629 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.01355311778597859270131890359, −5.76210118054371517262020054989, −5.67977854501064474690570106947, −5.40533470545678818905974101564, −5.36036177362494260385261017157, −4.73401512552929285281975518636, −4.71367160765961938200506968206, −4.70530082437166756509549457811, −4.67338120914734568974097030334, −4.53246181097811265902315628208, −3.96712978625600074749006692804, −3.86896209949477103890157631348, −3.64577480205658030495224175197, −3.62075680366259142733934451502, −3.56343859013510770513136402419, −3.24936797193772198072486306143, −3.00766383161831255895532087816, −2.77032007654021612627617461231, −2.30666109723907007123194318256, −2.14254364341070777381892870074, −1.81986020046948973544620214740, −1.38585558305090380304521523032, −1.28165043048159175423304649322, −1.18436371530917250460372208290, −1.11834923788114409681819649097, 0, 0, 0, 0, 0, 1.11834923788114409681819649097, 1.18436371530917250460372208290, 1.28165043048159175423304649322, 1.38585558305090380304521523032, 1.81986020046948973544620214740, 2.14254364341070777381892870074, 2.30666109723907007123194318256, 2.77032007654021612627617461231, 3.00766383161831255895532087816, 3.24936797193772198072486306143, 3.56343859013510770513136402419, 3.62075680366259142733934451502, 3.64577480205658030495224175197, 3.86896209949477103890157631348, 3.96712978625600074749006692804, 4.53246181097811265902315628208, 4.67338120914734568974097030334, 4.70530082437166756509549457811, 4.71367160765961938200506968206, 4.73401512552929285281975518636, 5.36036177362494260385261017157, 5.40533470545678818905974101564, 5.67977854501064474690570106947, 5.76210118054371517262020054989, 6.01355311778597859270131890359

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