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Properties

Label 10-3640e5-1.1-c1e5-0-1
Degree $10$
Conductor $6.390\times 10^{17}$
Sign $-1$
Analytic cond. $2.07439\times 10^{7}$
Root an. cond. $5.39124$
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  − 5·5-s + 5·7-s − 7·9-s − 7·11-s − 5·13-s + 17-s + 3·19-s − 5·23-s + 15·25-s − 3·27-s − 9·29-s + 6·31-s − 25·35-s − 10·37-s − 8·41-s + 6·43-s + 35·45-s − 13·47-s + 15·49-s − 16·53-s + 35·55-s − 59-s − 11·61-s − 35·63-s + 25·65-s − 30·67-s − 12·71-s + ⋯
L(s)  = 1  − 2.23·5-s + 1.88·7-s − 7/3·9-s − 2.11·11-s − 1.38·13-s + 0.242·17-s + 0.688·19-s − 1.04·23-s + 3·25-s − 0.577·27-s − 1.67·29-s + 1.07·31-s − 4.22·35-s − 1.64·37-s − 1.24·41-s + 0.914·43-s + 5.21·45-s − 1.89·47-s + 15/7·49-s − 2.19·53-s + 4.71·55-s − 0.130·59-s − 1.40·61-s − 4.40·63-s + 3.10·65-s − 3.66·67-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 5^{5} \cdot 7^{5} \cdot 13^{5}\)
Sign: $-1$
Analytic conductor: \(2.07439\times 10^{7}\)
Root analytic conductor: \(5.39124\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{15} \cdot 5^{5} \cdot 7^{5} \cdot 13^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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$C_1$ \( ( 1 + T )^{5} \)
7$C_1$ \( ( 1 - T )^{5} \)
13$C_1$ \( ( 1 + T )^{5} \)
good3$C_2\times (C_2^4 : D_5)$ \( 1 + 7 T^{2} + p T^{3} + 28 T^{4} + 14 T^{5} + 28 p T^{6} + p^{3} T^{7} + 7 p^{3} T^{8} + p^{5} T^{10} \)
11$C_2\times (C_2^4 : D_5)$ \( 1 + 7 T + 45 T^{2} + 224 T^{3} + 976 T^{4} + 3298 T^{5} + 976 p T^{6} + 224 p^{2} T^{7} + 45 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2\times (C_2^4 : D_5)$ \( 1 - T + 38 T^{2} - 122 T^{3} + 905 T^{4} - 2794 T^{5} + 905 p T^{6} - 122 p^{2} T^{7} + 38 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
19$C_2\times (C_2^4 : D_5)$ \( 1 - 3 T + 40 T^{2} - 68 T^{3} + 1195 T^{4} - 2586 T^{5} + 1195 p T^{6} - 68 p^{2} T^{7} + 40 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2\times (C_2^4 : D_5)$ \( 1 + 5 T + 45 T^{2} - 60 T^{3} - 124 T^{4} - 6722 T^{5} - 124 p T^{6} - 60 p^{2} T^{7} + 45 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2\times (C_2^4 : D_5)$ \( 1 + 9 T + 86 T^{2} + 462 T^{3} + 3929 T^{4} + 19826 T^{5} + 3929 p T^{6} + 462 p^{2} T^{7} + 86 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2\times (C_2^4 : D_5)$ \( 1 - 6 T + 73 T^{2} - 323 T^{3} + 3182 T^{4} - 11570 T^{5} + 3182 p T^{6} - 323 p^{2} T^{7} + 73 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2\times (C_2^4 : D_5)$ \( 1 + 10 T + 5 p T^{2} + 1435 T^{3} + 13666 T^{4} + 78830 T^{5} + 13666 p T^{6} + 1435 p^{2} T^{7} + 5 p^{4} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2\times (C_2^4 : D_5)$ \( 1 + 8 T + 115 T^{2} + 1053 T^{3} + 8336 T^{4} + 56110 T^{5} + 8336 p T^{6} + 1053 p^{2} T^{7} + 115 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2\times (C_2^4 : D_5)$ \( 1 - 6 T + 51 T^{2} - 64 T^{3} + 1126 T^{4} + 3212 T^{5} + 1126 p T^{6} - 64 p^{2} T^{7} + 51 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2\times (C_2^4 : D_5)$ \( 1 + 13 T + 219 T^{2} + 2080 T^{3} + 20506 T^{4} + 137830 T^{5} + 20506 p T^{6} + 2080 p^{2} T^{7} + 219 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2\times (C_2^4 : D_5)$ \( 1 + 16 T + 241 T^{2} + 2400 T^{3} + 23842 T^{4} + 177440 T^{5} + 23842 p T^{6} + 2400 p^{2} T^{7} + 241 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2\times (C_2^4 : D_5)$ \( 1 + T + 224 T^{2} + 6 T^{3} + 22431 T^{4} - 5854 T^{5} + 22431 p T^{6} + 6 p^{2} T^{7} + 224 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
61$C_2\times (C_2^4 : D_5)$ \( 1 + 11 T + 257 T^{2} + 1748 T^{3} + 24890 T^{4} + 127282 T^{5} + 24890 p T^{6} + 1748 p^{2} T^{7} + 257 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2\times (C_2^4 : D_5)$ \( 1 + 30 T + 581 T^{2} + 7619 T^{3} + 1220 p T^{4} + 713318 T^{5} + 1220 p^{2} T^{6} + 7619 p^{2} T^{7} + 581 p^{3} T^{8} + 30 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2\times (C_2^4 : D_5)$ \( 1 + 12 T + 351 T^{2} + 2944 T^{3} + 48102 T^{4} + 296424 T^{5} + 48102 p T^{6} + 2944 p^{2} T^{7} + 351 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2\times (C_2^4 : D_5)$ \( 1 - 23 T + 401 T^{2} - 5008 T^{3} + 54966 T^{4} - 487250 T^{5} + 54966 p T^{6} - 5008 p^{2} T^{7} + 401 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2\times (C_2^4 : D_5)$ \( 1 - 2 T + 149 T^{2} - 325 T^{3} + 6144 T^{4} - 25058 T^{5} + 6144 p T^{6} - 325 p^{2} T^{7} + 149 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2\times (C_2^4 : D_5)$ \( 1 + 2 T + 203 T^{2} - 496 T^{3} + 17894 T^{4} - 107460 T^{5} + 17894 p T^{6} - 496 p^{2} T^{7} + 203 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2\times (C_2^4 : D_5)$ \( 1 + 25 T + 490 T^{2} + 5310 T^{3} + 55781 T^{4} + 448754 T^{5} + 55781 p T^{6} + 5310 p^{2} T^{7} + 490 p^{3} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2\times (C_2^4 : D_5)$ \( 1 + 5 T + 367 T^{2} + 1652 T^{3} + 63232 T^{4} + 223022 T^{5} + 63232 p T^{6} + 1652 p^{2} T^{7} + 367 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.36274315641329399006502804941, −5.12793516326234278790497335556, −5.10193961099023758894024139219, −5.01317339926538442264785952417, −4.86002039266486887074775415955, −4.80414553659645810738698178801, −4.58707075671560876434763350611, −4.31430885990216571924302405343, −4.12591475985970370471108407296, −4.10292478336619107641313114002, −3.72881724618049105330127013254, −3.56280013296045319663965793437, −3.42164879884499011076632028910, −3.34094770951071722910562684366, −3.17474189809578312520743219721, −2.80047521280895651640945899625, −2.58325519144815968624140872546, −2.55634005837277883175771090461, −2.54418167843084901512021830060, −2.37504081991936772784075707998, −1.79061753565348062261397577800, −1.65066095981150055762927947751, −1.48786643616585563309624968970, −1.18576167267202214895480562939, −1.10219643235048271476675755032, 0, 0, 0, 0, 0, 1.10219643235048271476675755032, 1.18576167267202214895480562939, 1.48786643616585563309624968970, 1.65066095981150055762927947751, 1.79061753565348062261397577800, 2.37504081991936772784075707998, 2.54418167843084901512021830060, 2.55634005837277883175771090461, 2.58325519144815968624140872546, 2.80047521280895651640945899625, 3.17474189809578312520743219721, 3.34094770951071722910562684366, 3.42164879884499011076632028910, 3.56280013296045319663965793437, 3.72881724618049105330127013254, 4.10292478336619107641313114002, 4.12591475985970370471108407296, 4.31430885990216571924302405343, 4.58707075671560876434763350611, 4.80414553659645810738698178801, 4.86002039266486887074775415955, 5.01317339926538442264785952417, 5.10193961099023758894024139219, 5.12793516326234278790497335556, 5.36274315641329399006502804941

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