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Properties

Label 10-4600e5-1.1-c1e5-0-3
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 $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 6·9-s − 4·13-s − 6·17-s + 5·23-s − 3·27-s + 12·29-s − 18·31-s − 10·37-s − 6·41-s − 10·43-s − 22·47-s − 2·49-s − 10·53-s − 59-s + 10·61-s + 24·63-s − 8·67-s + 8·71-s − 6·73-s + 9·81-s + 2·83-s + 14·89-s + 16·91-s − 6·97-s − 3·101-s − 2·103-s + ⋯
L(s)  = 1  − 1.51·7-s − 2·9-s − 1.10·13-s − 1.45·17-s + 1.04·23-s − 0.577·27-s + 2.22·29-s − 3.23·31-s − 1.64·37-s − 0.937·41-s − 1.52·43-s − 3.20·47-s − 2/7·49-s − 1.37·53-s − 0.130·59-s + 1.28·61-s + 3.02·63-s − 0.977·67-s + 0.949·71-s − 0.702·73-s + 81-s + 0.219·83-s + 1.48·89-s + 1.67·91-s − 0.609·97-s − 0.298·101-s − 0.197·103-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{15} \cdot 5^{10} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + 2 p T^{2} + p T^{3} + p^{3} T^{4} + 2 p T^{5} + p^{4} T^{6} + p^{3} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 4 T + 18 T^{2} + 36 T^{3} + 153 T^{4} + 312 T^{5} + 153 p T^{6} + 36 p^{2} T^{7} + 18 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 40 T^{2} - 6 T^{3} + 763 T^{4} - 108 T^{5} + 763 p T^{6} - 6 p^{2} T^{7} + 40 p^{3} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 33 T^{2} + 9 p T^{3} + 729 T^{4} + 2037 T^{5} + 729 p T^{6} + 9 p^{3} T^{7} + 33 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 6 T + 37 T^{2} + 48 T^{3} - 230 T^{4} - 2220 T^{5} - 230 p T^{6} + 48 p^{2} T^{7} + 37 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 80 T^{2} - 6 T^{3} + 2803 T^{4} - 204 T^{5} + 2803 p T^{6} - 6 p^{2} T^{7} + 80 p^{3} T^{8} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 12 T + 5 p T^{2} - 1053 T^{3} + 8137 T^{4} - 43677 T^{5} + 8137 p T^{6} - 1053 p^{2} T^{7} + 5 p^{4} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 18 T + 8 p T^{2} + 2277 T^{3} + 17743 T^{4} + 106350 T^{5} + 17743 p T^{6} + 2277 p^{2} T^{7} + 8 p^{4} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 10 T + 153 T^{2} + 984 T^{3} + 9210 T^{4} + 46044 T^{5} + 9210 p T^{6} + 984 p^{2} T^{7} + 153 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 6 T + 139 T^{2} + 747 T^{3} + 10003 T^{4} + 41535 T^{5} + 10003 p T^{6} + 747 p^{2} T^{7} + 139 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 10 T + 192 T^{2} + 1368 T^{3} + 15411 T^{4} + 82140 T^{5} + 15411 p T^{6} + 1368 p^{2} T^{7} + 192 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 22 T + 332 T^{2} + 3835 T^{3} + 35433 T^{4} + 263290 T^{5} + 35433 p T^{6} + 3835 p^{2} T^{7} + 332 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 10 T + 173 T^{2} + 1720 T^{3} + 14166 T^{4} + 125884 T^{5} + 14166 p T^{6} + 1720 p^{2} T^{7} + 173 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + T + 212 T^{2} + 343 T^{3} + 21441 T^{4} + 30508 T^{5} + 21441 p T^{6} + 343 p^{2} T^{7} + 212 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 10 T + 105 T^{2} - 1368 T^{3} + 13410 T^{4} - 84828 T^{5} + 13410 p T^{6} - 1368 p^{2} T^{7} + 105 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 8 T + 147 T^{2} - 264 T^{3} - 1602 T^{4} - 116928 T^{5} - 1602 p T^{6} - 264 p^{2} T^{7} + 147 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 8 T + 176 T^{2} - 1583 T^{3} + 18243 T^{4} - 136046 T^{5} + 18243 p T^{6} - 1583 p^{2} T^{7} + 176 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 6 T + 143 T^{2} + 1713 T^{3} + 15079 T^{4} + 157581 T^{5} + 15079 p T^{6} + 1713 p^{2} T^{7} + 143 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 242 T^{2} - 36 T^{3} + 31333 T^{4} - 7416 T^{5} + 31333 p T^{6} - 36 p^{2} T^{7} + 242 p^{3} T^{8} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 2 T + 224 T^{2} + 448 T^{3} + 21855 T^{4} + 99532 T^{5} + 21855 p T^{6} + 448 p^{2} T^{7} + 224 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 14 T + 401 T^{2} - 3704 T^{3} + 62598 T^{4} - 434420 T^{5} + 62598 p T^{6} - 3704 p^{2} T^{7} + 401 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 6 T + 341 T^{2} + 1512 T^{3} + 52762 T^{4} + 180324 T^{5} + 52762 p T^{6} + 1512 p^{2} T^{7} + 341 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.20307984777574094229306324949, −5.09378947524076406896767298661, −5.04046188139164486141686633439, −4.90857028334821358683371695760, −4.86691070869487248317629918308, −4.58365262694259645078219343723, −4.43076328522899383664274054392, −4.15459887210032792754795048357, −3.88100978729805592850675984829, −3.84185520472874140010348793103, −3.42197564556176660376842552779, −3.41268378351922524326574516864, −3.37873314373131666408858326814, −3.27365857627806664241381909040, −3.17068458023189114495646357554, −2.69590541641061550550250299404, −2.58542720414934539616655177691, −2.47700239908320072111534266682, −2.37484328509261378529483379612, −2.19439798856536551129031234110, −1.86413073333777906906533203244, −1.51330792393091999993441701124, −1.39850697818156706980006876849, −1.32134206307823997720091627963, −0.947566975432929896263274120837, 0, 0, 0, 0, 0, 0.947566975432929896263274120837, 1.32134206307823997720091627963, 1.39850697818156706980006876849, 1.51330792393091999993441701124, 1.86413073333777906906533203244, 2.19439798856536551129031234110, 2.37484328509261378529483379612, 2.47700239908320072111534266682, 2.58542720414934539616655177691, 2.69590541641061550550250299404, 3.17068458023189114495646357554, 3.27365857627806664241381909040, 3.37873314373131666408858326814, 3.41268378351922524326574516864, 3.42197564556176660376842552779, 3.84185520472874140010348793103, 3.88100978729805592850675984829, 4.15459887210032792754795048357, 4.43076328522899383664274054392, 4.58365262694259645078219343723, 4.86691070869487248317629918308, 4.90857028334821358683371695760, 5.04046188139164486141686633439, 5.09378947524076406896767298661, 5.20307984777574094229306324949

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