Properties

Label 10-4600e5-1.1-c1e5-0-1
Degree $10$
Conductor $2.060\times 10^{18}$
Sign $1$
Analytic cond. $6.68612\times 10^{7}$
Root an. cond. $6.06062$
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  + 3·3-s − 7-s − 9-s − 4·11-s − 13-s + 5·17-s + 4·19-s − 3·21-s + 5·23-s − 13·27-s − 11·29-s + 4·31-s − 12·33-s + 6·37-s − 3·39-s − 8·41-s + 3·43-s + 2·47-s − 18·49-s + 15·51-s + 18·53-s + 12·57-s + 23·59-s − 26·61-s + 63-s + 3·67-s + 15·69-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.377·7-s − 1/3·9-s − 1.20·11-s − 0.277·13-s + 1.21·17-s + 0.917·19-s − 0.654·21-s + 1.04·23-s − 2.50·27-s − 2.04·29-s + 0.718·31-s − 2.08·33-s + 0.986·37-s − 0.480·39-s − 1.24·41-s + 0.457·43-s + 0.291·47-s − 2.57·49-s + 2.10·51-s + 2.47·53-s + 1.58·57-s + 2.99·59-s − 3.32·61-s + 0.125·63-s + 0.366·67-s + 1.80·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 5^{10} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(6.68612\times 10^{7}\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{15} \cdot 5^{10} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.20136701\)
\(L(\frac12)\) \(\approx\) \(13.20136701\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 - p T + 10 T^{2} - 20 T^{3} + 44 T^{4} - 76 T^{5} + 44 p T^{6} - 20 p^{2} T^{7} + 10 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + T + 19 T^{2} + 16 T^{3} + 30 p T^{4} + 158 T^{5} + 30 p^{2} T^{6} + 16 p^{2} T^{7} + 19 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 4 T + 38 T^{2} + 8 p T^{3} + 557 T^{4} + 960 T^{5} + 557 p T^{6} + 8 p^{3} T^{7} + 38 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + T + 28 T^{2} + 4 T^{3} + 568 T^{4} + 220 T^{5} + 568 p T^{6} + 4 p^{2} T^{7} + 28 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 5 T + 57 T^{2} - 224 T^{3} + 1390 T^{4} - 4758 T^{5} + 1390 p T^{6} - 224 p^{2} T^{7} + 57 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 42 T^{2} - 40 T^{3} + 317 T^{4} + 1304 T^{5} + 317 p T^{6} - 40 p^{2} T^{7} + 42 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 11 T + 134 T^{2} + 970 T^{3} + 238 p T^{4} + 37502 T^{5} + 238 p^{2} T^{6} + 970 p^{2} T^{7} + 134 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 4 T + 80 T^{2} - 337 T^{3} + 3437 T^{4} - 12806 T^{5} + 3437 p T^{6} - 337 p^{2} T^{7} + 80 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 6 T + 45 T^{2} - 312 T^{3} + 2822 T^{4} - 12004 T^{5} + 2822 p T^{6} - 312 p^{2} T^{7} + 45 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 8 T + 153 T^{2} + 641 T^{3} + 8317 T^{4} + 23597 T^{5} + 8317 p T^{6} + 641 p^{2} T^{7} + 153 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 3 T + 71 T^{2} - 192 T^{3} + 4450 T^{4} - 13194 T^{5} + 4450 p T^{6} - 192 p^{2} T^{7} + 71 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 2 T + 68 T^{2} - 343 T^{3} + 3101 T^{4} - 29318 T^{5} + 3101 p T^{6} - 343 p^{2} T^{7} + 68 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 18 T + 213 T^{2} - 1184 T^{3} + 4462 T^{4} - 2236 T^{5} + 4462 p T^{6} - 1184 p^{2} T^{7} + 213 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 23 T + 348 T^{2} - 4341 T^{3} + 43633 T^{4} - 357240 T^{5} + 43633 p T^{6} - 4341 p^{2} T^{7} + 348 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 26 T + 541 T^{2} + 7192 T^{3} + 81278 T^{4} + 683676 T^{5} + 81278 p T^{6} + 7192 p^{2} T^{7} + 541 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 3 T + 275 T^{2} - 612 T^{3} + 33150 T^{4} - 55586 T^{5} + 33150 p T^{6} - 612 p^{2} T^{7} + 275 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 2 T + 288 T^{2} + 435 T^{3} + 36805 T^{4} + 41598 T^{5} + 36805 p T^{6} + 435 p^{2} T^{7} + 288 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 4 T - 3 T^{2} - 167 T^{3} + 6489 T^{4} - 34251 T^{5} + 6489 p T^{6} - 167 p^{2} T^{7} - 3 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 43 T + 1029 T^{2} - 16880 T^{3} + 210588 T^{4} - 2084442 T^{5} + 210588 p T^{6} - 16880 p^{2} T^{7} + 1029 p^{3} T^{8} - 43 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 30 T + 584 T^{2} - 8400 T^{3} + 102155 T^{4} - 1014020 T^{5} + 102155 p T^{6} - 8400 p^{2} T^{7} + 584 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 15 T + 291 T^{2} - 2152 T^{3} + 25484 T^{4} - 1554 p T^{5} + 25484 p T^{6} - 2152 p^{2} T^{7} + 291 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 8 T + 257 T^{2} - 1592 T^{3} + 32590 T^{4} - 159328 T^{5} + 32590 p T^{6} - 1592 p^{2} T^{7} + 257 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.92107632250966524180419680410, −4.78018421181576912275044699765, −4.73481987582282592891317122046, −4.35465190396408181442373464583, −4.14640688511784160713609313731, −4.14630040281573244121365584044, −3.64345817891646406450966279031, −3.62619269274518203991227989468, −3.58725029922366523302900208259, −3.43420401595763752559117758396, −3.12948556148899701169402683173, −3.12038167984129461936910417998, −2.91277295066977399845641036854, −2.72748324374623052125566903521, −2.71128488763579903998468761077, −2.41740529786159430444301270108, −2.01576515114647906952004263852, −1.98884649052487709158668838898, −1.91293555914909902824839103715, −1.85680878524738529467615954208, −1.25859972964779520839848282471, −0.976781587510391934631316268951, −0.66519017867005042611750514233, −0.50824252168474403704510750921, −0.45389790267112783386339757269, 0.45389790267112783386339757269, 0.50824252168474403704510750921, 0.66519017867005042611750514233, 0.976781587510391934631316268951, 1.25859972964779520839848282471, 1.85680878524738529467615954208, 1.91293555914909902824839103715, 1.98884649052487709158668838898, 2.01576515114647906952004263852, 2.41740529786159430444301270108, 2.71128488763579903998468761077, 2.72748324374623052125566903521, 2.91277295066977399845641036854, 3.12038167984129461936910417998, 3.12948556148899701169402683173, 3.43420401595763752559117758396, 3.58725029922366523302900208259, 3.62619269274518203991227989468, 3.64345817891646406450966279031, 4.14630040281573244121365584044, 4.14640688511784160713609313731, 4.35465190396408181442373464583, 4.73481987582282592891317122046, 4.78018421181576912275044699765, 4.92107632250966524180419680410

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