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Properties

Label 10-1911e5-1.1-c1e5-0-0
Degree $10$
Conductor $2.549\times 10^{16}$
Sign $1$
Analytic cond. $827349.$
Root an. cond. $3.90632$
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  + 2·2-s − 5·3-s − 3·5-s − 10·6-s − 2·8-s + 15·9-s − 6·10-s + 6·11-s − 5·13-s + 15·15-s − 3·16-s + 10·17-s + 30·18-s − 5·19-s + 12·22-s + 23-s + 10·24-s − 10·26-s − 35·27-s + 17·29-s + 30·30-s − 31-s − 4·32-s − 30·33-s + 20·34-s + 4·37-s − 10·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.88·3-s − 1.34·5-s − 4.08·6-s − 0.707·8-s + 5·9-s − 1.89·10-s + 1.80·11-s − 1.38·13-s + 3.87·15-s − 3/4·16-s + 2.42·17-s + 7.07·18-s − 1.14·19-s + 2.55·22-s + 0.208·23-s + 2.04·24-s − 1.96·26-s − 6.73·27-s + 3.15·29-s + 5.47·30-s − 0.179·31-s − 0.707·32-s − 5.22·33-s + 3.42·34-s + 0.657·37-s − 1.62·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 7^{10} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 7^{10} \cdot 13^{5}\)
Sign: $1$
Analytic conductor: \(827349.\)
Root analytic conductor: \(3.90632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 7^{10} \cdot 13^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.109609294\)
\(L(\frac12)\) \(\approx\) \(3.109609294\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{5} \)
7 \( 1 \)
13$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - p T + p^{2} T^{2} - 3 p T^{3} + 11 T^{4} - p^{4} T^{5} + 11 p T^{6} - 3 p^{3} T^{7} + p^{5} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 3 T + 9 T^{2} + 4 p T^{3} + 6 p T^{4} + 54 T^{5} + 6 p^{2} T^{6} + 4 p^{3} T^{7} + 9 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 6 T + 39 T^{2} - 128 T^{3} + 502 T^{4} - 1372 T^{5} + 502 p T^{6} - 128 p^{2} T^{7} + 39 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 10 T + 53 T^{2} - 232 T^{3} + 1274 T^{4} - 5980 T^{5} + 1274 p T^{6} - 232 p^{2} T^{7} + 53 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 5 T + 39 T^{2} - 36 T^{3} - 286 T^{4} - 5042 T^{5} - 286 p T^{6} - 36 p^{2} T^{7} + 39 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - T + 55 T^{2} - 116 T^{3} + 1694 T^{4} - 4694 T^{5} + 1694 p T^{6} - 116 p^{2} T^{7} + 55 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 17 T + 177 T^{2} - 1228 T^{3} + 242 p T^{4} - 36566 T^{5} + 242 p^{2} T^{6} - 1228 p^{2} T^{7} + 177 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + T + 107 T^{2} + 92 T^{3} + 5402 T^{4} + 3846 T^{5} + 5402 p T^{6} + 92 p^{2} T^{7} + 107 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 97 T^{2} - 144 T^{3} + 3490 T^{4} + 360 T^{5} + 3490 p T^{6} - 144 p^{2} T^{7} + 97 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 14 T + 173 T^{2} + 1648 T^{3} + 13862 T^{4} + 90956 T^{5} + 13862 p T^{6} + 1648 p^{2} T^{7} + 173 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + T + 143 T^{2} + 76 T^{3} + 10098 T^{4} + 4630 T^{5} + 10098 p T^{6} + 76 p^{2} T^{7} + 143 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 3 T + 63 T^{2} + 164 T^{3} + 5162 T^{4} - 8886 T^{5} + 5162 p T^{6} + 164 p^{2} T^{7} + 63 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 17 T + 281 T^{2} - 2988 T^{3} + 29466 T^{4} - 225334 T^{5} + 29466 p T^{6} - 2988 p^{2} T^{7} + 281 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 8 T + 111 T^{2} - 736 T^{3} + 10894 T^{4} - 65984 T^{5} + 10894 p T^{6} - 736 p^{2} T^{7} + 111 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 18 T + 313 T^{2} - 3448 T^{3} + 37090 T^{4} - 295308 T^{5} + 37090 p T^{6} - 3448 p^{2} T^{7} + 313 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 16 T + 319 T^{2} - 3456 T^{3} + 41290 T^{4} - 330720 T^{5} + 41290 p T^{6} - 3456 p^{2} T^{7} + 319 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 16 T + 259 T^{2} + 2408 T^{3} + 24806 T^{4} + 179216 T^{5} + 24806 p T^{6} + 2408 p^{2} T^{7} + 259 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 23 T + 525 T^{2} + 6964 T^{3} + 87738 T^{4} + 771786 T^{5} + 87738 p T^{6} + 6964 p^{2} T^{7} + 525 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 3 T + 163 T^{2} - 20 p T^{3} + 14482 T^{4} - 186258 T^{5} + 14482 p T^{6} - 20 p^{3} T^{7} + 163 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 11 T + 331 T^{2} - 3612 T^{3} + 48066 T^{4} - 448086 T^{5} + 48066 p T^{6} - 3612 p^{2} T^{7} + 331 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 35 T + 789 T^{2} - 12884 T^{3} + 167254 T^{4} - 1734086 T^{5} + 167254 p T^{6} - 12884 p^{2} T^{7} + 789 p^{3} T^{8} - 35 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 27 T + 453 T^{2} + 5204 T^{3} + 49498 T^{4} + 446498 T^{5} + 49498 p T^{6} + 5204 p^{2} T^{7} + 453 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.31974523404287702614896668634, −5.31217109596806587656075531398, −5.24182391440259403281374755573, −5.01001199934170250875267615732, −4.85138289882651664582188220936, −4.45550891642839627997309916182, −4.41772053390734449629283548798, −4.37219963921954291633364115099, −4.32283200779366670088237393593, −4.23133084320399953753082400746, −3.67943120775441208670342152324, −3.63217515785935391636592938882, −3.56652493087685920462495253200, −3.47532105435438818330935910024, −2.98370999149578798563000999628, −2.79814166847247100068046144426, −2.59020515808604448381284659391, −2.20288203940855605129089857032, −1.84856617627276058254069744879, −1.80837112762226878128633224899, −1.39239202377663597195958827070, −1.01260543720156827210262163289, −0.68884820551918585002336497459, −0.56196759167419549044524044573, −0.51434248974035396821620010458, 0.51434248974035396821620010458, 0.56196759167419549044524044573, 0.68884820551918585002336497459, 1.01260543720156827210262163289, 1.39239202377663597195958827070, 1.80837112762226878128633224899, 1.84856617627276058254069744879, 2.20288203940855605129089857032, 2.59020515808604448381284659391, 2.79814166847247100068046144426, 2.98370999149578798563000999628, 3.47532105435438818330935910024, 3.56652493087685920462495253200, 3.63217515785935391636592938882, 3.67943120775441208670342152324, 4.23133084320399953753082400746, 4.32283200779366670088237393593, 4.37219963921954291633364115099, 4.41772053390734449629283548798, 4.45550891642839627997309916182, 4.85138289882651664582188220936, 5.01001199934170250875267615732, 5.24182391440259403281374755573, 5.31217109596806587656075531398, 5.31974523404287702614896668634

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