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Properties

Label 10-1155e5-1.1-c1e5-0-0
Degree $10$
Conductor $2.055\times 10^{15}$
Sign $1$
Analytic cond. $66726.0$
Root an. cond. $3.03689$
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-s + 5·3-s − 5·5-s + 5·6-s + 5·7-s + 15·9-s − 5·10-s + 5·11-s + 8·13-s + 5·14-s − 25·15-s − 16-s + 15·18-s + 6·19-s + 25·21-s + 5·22-s − 2·23-s + 15·25-s + 8·26-s + 35·27-s + 6·29-s − 25·30-s + 10·31-s − 32-s + 25·33-s − 25·35-s + 4·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.88·3-s − 2.23·5-s + 2.04·6-s + 1.88·7-s + 5·9-s − 1.58·10-s + 1.50·11-s + 2.21·13-s + 1.33·14-s − 6.45·15-s − 1/4·16-s + 3.53·18-s + 1.37·19-s + 5.45·21-s + 1.06·22-s − 0.417·23-s + 3·25-s + 1.56·26-s + 6.73·27-s + 1.11·29-s − 4.56·30-s + 1.79·31-s − 0.176·32-s + 4.35·33-s − 4.22·35-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 7^{5} \cdot 11^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 5^{5} \cdot 7^{5} \cdot 11^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{5} \cdot 5^{5} \cdot 7^{5} \cdot 11^{5}\)
Sign: $1$
Analytic conductor: \(66726.0\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 3^{5} \cdot 5^{5} \cdot 7^{5} \cdot 11^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(40.04714902\)
\(L(\frac12)\) \(\approx\) \(40.04714902\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{5} \)
7$C_1$ \( ( 1 - T )^{5} \)
11$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - T + T^{2} - T^{3} + p T^{4} - p T^{5} + p^{2} T^{6} - p^{2} T^{7} + p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 8 T + 55 T^{2} - 252 T^{3} + 1260 T^{4} - 4632 T^{5} + 1260 p T^{6} - 252 p^{2} T^{7} + 55 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 22 T^{2} + 70 T^{3} + 57 T^{4} + 1876 T^{5} + 57 p T^{6} + 70 p^{2} T^{7} + 22 p^{3} T^{8} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 92 T^{2} - 412 T^{3} + 181 p T^{4} - 11388 T^{5} + 181 p^{2} T^{6} - 412 p^{2} T^{7} + 92 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 2 T + 68 T^{2} + 260 T^{3} + 2175 T^{4} + 9652 T^{5} + 2175 p T^{6} + 260 p^{2} T^{7} + 68 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 6 T + 2 p T^{2} - 142 T^{3} + 1365 T^{4} - 104 p T^{5} + 1365 p T^{6} - 142 p^{2} T^{7} + 2 p^{4} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 10 T + 145 T^{2} - 1040 T^{3} + 8424 T^{4} - 45388 T^{5} + 8424 p T^{6} - 1040 p^{2} T^{7} + 145 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 71 T^{2} - 204 T^{3} + 1316 T^{4} - 4928 T^{5} + 1316 p T^{6} - 204 p^{2} T^{7} + 71 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 83 T^{2} - 260 T^{3} + 4964 T^{4} - 11192 T^{5} + 4964 p T^{6} - 260 p^{2} T^{7} + 83 p^{3} T^{8} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 48 T^{2} + 12 T^{3} + 2563 T^{4} + 12104 T^{5} + 2563 p T^{6} + 12 p^{2} T^{7} + 48 p^{3} T^{8} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 2 T + 97 T^{2} + 328 T^{3} + 6984 T^{4} + 12780 T^{5} + 6984 p T^{6} + 328 p^{2} T^{7} + 97 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 8 T + 194 T^{2} + 1338 T^{3} + 17981 T^{4} + 100460 T^{5} + 17981 p T^{6} + 1338 p^{2} T^{7} + 194 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 4 T + 164 T^{2} + 404 T^{3} + 15719 T^{4} + 36528 T^{5} + 15719 p T^{6} + 404 p^{2} T^{7} + 164 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 246 T^{2} - 2642 T^{3} + 28105 T^{4} - 220396 T^{5} + 28105 p T^{6} - 2642 p^{2} T^{7} + 246 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 16 T + 323 T^{2} - 3792 T^{3} + 43246 T^{4} - 364224 T^{5} + 43246 p T^{6} - 3792 p^{2} T^{7} + 323 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 6 T + 161 T^{2} + 392 T^{3} + 13184 T^{4} + 19748 T^{5} + 13184 p T^{6} + 392 p^{2} T^{7} + 161 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 2 T + 121 T^{2} + 240 T^{3} + 13998 T^{4} - 10908 T^{5} + 13998 p T^{6} + 240 p^{2} T^{7} + 121 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 42 T + 1065 T^{2} - 18336 T^{3} + 239440 T^{4} - 2398060 T^{5} + 239440 p T^{6} - 18336 p^{2} T^{7} + 1065 p^{3} T^{8} - 42 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 22 T + 432 T^{2} + 5200 T^{3} + 64115 T^{4} + 577172 T^{5} + 64115 p T^{6} + 5200 p^{2} T^{7} + 432 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 10 T + 202 T^{2} + 1822 T^{3} + 26405 T^{4} + 244592 T^{5} + 26405 p T^{6} + 1822 p^{2} T^{7} + 202 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 2 T + 58 T^{2} + 502 T^{3} + 12309 T^{4} + 560 T^{5} + 12309 p T^{6} + 502 p^{2} T^{7} + 58 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.09690868727350506185242526080, −5.46399584260304120802708049630, −5.42087465506184896421657937287, −5.35140043162231701887602198012, −5.12701870544903204030314817238, −4.61837879106746725480508806681, −4.48525786874413755529092555832, −4.46165381442901465434453013402, −4.37737206667780255461239884531, −4.37520801385368843504160525730, −3.82660749712149201864831250705, −3.72535845293637296989215737776, −3.59360221830395648810042673571, −3.58093298988371494450758179629, −3.26856566858543446774365480886, −3.06507203358733885427050255801, −2.82136079909335777599720043105, −2.66874038196371146122925188530, −2.21103285528245149504466251361, −2.01525763931295709078079235613, −1.85852278820446617148957629358, −1.42619115367639981044867856825, −1.01025351930272324010087641915, −0.958313057404705504034689164102, −0.904439964223657288291777772045, 0.904439964223657288291777772045, 0.958313057404705504034689164102, 1.01025351930272324010087641915, 1.42619115367639981044867856825, 1.85852278820446617148957629358, 2.01525763931295709078079235613, 2.21103285528245149504466251361, 2.66874038196371146122925188530, 2.82136079909335777599720043105, 3.06507203358733885427050255801, 3.26856566858543446774365480886, 3.58093298988371494450758179629, 3.59360221830395648810042673571, 3.72535845293637296989215737776, 3.82660749712149201864831250705, 4.37520801385368843504160525730, 4.37737206667780255461239884531, 4.46165381442901465434453013402, 4.48525786874413755529092555832, 4.61837879106746725480508806681, 5.12701870544903204030314817238, 5.35140043162231701887602198012, 5.42087465506184896421657937287, 5.46399584260304120802708049630, 6.09690868727350506185242526080

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