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Properties

Label 10-4025e5-1.1-c1e5-0-1
Degree $10$
Conductor $1.056\times 10^{18}$
Sign $1$
Analytic cond. $3.42937\times 10^{7}$
Root an. cond. $5.66919$
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  + 3·2-s + 6·3-s + 4-s + 18·6-s + 5·7-s − 7·8-s + 13·9-s − 11·11-s + 6·12-s + 7·13-s + 15·14-s − 11·16-s − 17-s + 39·18-s + 2·19-s + 30·21-s − 33·22-s − 5·23-s − 42·24-s + 21·26-s + 5·27-s + 5·28-s − 14·29-s − 6·31-s − 2·32-s − 66·33-s − 3·34-s + ⋯
L(s)  = 1  + 2.12·2-s + 3.46·3-s + 1/2·4-s + 7.34·6-s + 1.88·7-s − 2.47·8-s + 13/3·9-s − 3.31·11-s + 1.73·12-s + 1.94·13-s + 4.00·14-s − 2.75·16-s − 0.242·17-s + 9.19·18-s + 0.458·19-s + 6.54·21-s − 7.03·22-s − 1.04·23-s − 8.57·24-s + 4.11·26-s + 0.962·27-s + 0.944·28-s − 2.59·29-s − 1.07·31-s − 0.353·32-s − 11.4·33-s − 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{10} \cdot 7^{5} \cdot 23^{5}\)
Sign: $1$
Analytic conductor: \(3.42937\times 10^{7}\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 5^{10} \cdot 7^{5} \cdot 23^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(47.71060597\)
\(L(\frac12)\) \(\approx\) \(47.71060597\)
\(L(\frac{3}{2})\) 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)$
bad5 \( 1 \)
7$C_1$ \( ( 1 - T )^{5} \)
23$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - 3 T + p^{3} T^{2} - 7 p T^{3} + 3 p^{3} T^{4} - 33 T^{5} + 3 p^{4} T^{6} - 7 p^{3} T^{7} + p^{6} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 2 p T + 23 T^{2} - 65 T^{3} + 149 T^{4} - 283 T^{5} + 149 p T^{6} - 65 p^{2} T^{7} + 23 p^{3} T^{8} - 2 p^{5} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + p T + 69 T^{2} + 299 T^{3} + 1060 T^{4} + 3464 T^{5} + 1060 p T^{6} + 299 p^{2} T^{7} + 69 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 7 T + 50 T^{2} - 218 T^{3} + 1149 T^{4} - 4063 T^{5} + 1149 p T^{6} - 218 p^{2} T^{7} + 50 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + T + 25 T^{2} + 83 T^{3} + 704 T^{4} + 800 T^{5} + 704 p T^{6} + 83 p^{2} T^{7} + 25 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 2 T + 50 T^{2} + T^{3} + 1137 T^{4} + 1030 T^{5} + 1137 p T^{6} + p^{2} T^{7} + 50 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 14 T + 106 T^{2} + 324 T^{3} - 723 T^{4} - 12204 T^{5} - 723 p T^{6} + 324 p^{2} T^{7} + 106 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 6 T + 85 T^{2} + 479 T^{3} + 3931 T^{4} + 20655 T^{5} + 3931 p T^{6} + 479 p^{2} T^{7} + 85 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - T + 17 T^{2} + 11 T^{3} + 376 T^{4} - 10076 T^{5} + 376 p T^{6} + 11 p^{2} T^{7} + 17 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 7 T + 135 T^{2} - 1002 T^{3} + 9005 T^{4} - 58961 T^{5} + 9005 p T^{6} - 1002 p^{2} T^{7} + 135 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 12 T + 138 T^{2} - 1233 T^{3} + 10559 T^{4} - 69898 T^{5} + 10559 p T^{6} - 1233 p^{2} T^{7} + 138 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 24 T + 431 T^{2} - 5125 T^{3} + 50209 T^{4} - 375429 T^{5} + 50209 p T^{6} - 5125 p^{2} T^{7} + 431 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 21 T + 407 T^{2} - 4757 T^{3} + 50566 T^{4} - 385716 T^{5} + 50566 p T^{6} - 4757 p^{2} T^{7} + 407 p^{3} T^{8} - 21 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + T + 225 T^{2} + 105 T^{3} + 22792 T^{4} + 5604 T^{5} + 22792 p T^{6} + 105 p^{2} T^{7} + 225 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 7 T + 259 T^{2} - 1417 T^{3} + 29142 T^{4} - 123032 T^{5} + 29142 p T^{6} - 1417 p^{2} T^{7} + 259 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 35 T + 785 T^{2} - 11891 T^{3} + 140386 T^{4} - 1278568 T^{5} + 140386 p T^{6} - 11891 p^{2} T^{7} + 785 p^{3} T^{8} - 35 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 32 T + 619 T^{2} + 8265 T^{3} + 88661 T^{4} + 796645 T^{5} + 88661 p T^{6} + 8265 p^{2} T^{7} + 619 p^{3} T^{8} + 32 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 7 T + 276 T^{2} - 984 T^{3} + 30101 T^{4} - 64761 T^{5} + 30101 p T^{6} - 984 p^{2} T^{7} + 276 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 18 T + 376 T^{2} - 4711 T^{3} + 57271 T^{4} - 530898 T^{5} + 57271 p T^{6} - 4711 p^{2} T^{7} + 376 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - T + 211 T^{2} - 561 T^{3} + 26112 T^{4} - 69144 T^{5} + 26112 p T^{6} - 561 p^{2} T^{7} + 211 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - T + 215 T^{2} + 449 T^{3} + 25370 T^{4} + 79560 T^{5} + 25370 p T^{6} + 449 p^{2} T^{7} + 215 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 9 T + 214 T^{2} - 2015 T^{3} + 19163 T^{4} - 226496 T^{5} + 19163 p T^{6} - 2015 p^{2} T^{7} + 214 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.02567788839632063552240878289, −4.62749405665831549897715318354, −4.57489138718291226465089263430, −4.55762298474706677696697097271, −4.40054040388122085218212097179, −3.92046411490126000369589655410, −3.87319799182977153118946223094, −3.80706272808054249979105609226, −3.72336615249092472991978150838, −3.70178779394171200360642346684, −3.42884314800694963558913232865, −3.09065626108276218281096289573, −3.02626542695282840096341419691, −2.96907015561519263112131630385, −2.44485299346754833530220528690, −2.40422403611834824762936835456, −2.36353536965567212203166922228, −2.22540941458293592365041621615, −2.14988440555000381679971222649, −1.86365243900514391416274919269, −1.71628218759745979538909328781, −0.972131520870796013393978200253, −0.921074461841150025610731354074, −0.64720077622028406957842868682, −0.34324500732780793920264370012, 0.34324500732780793920264370012, 0.64720077622028406957842868682, 0.921074461841150025610731354074, 0.972131520870796013393978200253, 1.71628218759745979538909328781, 1.86365243900514391416274919269, 2.14988440555000381679971222649, 2.22540941458293592365041621615, 2.36353536965567212203166922228, 2.40422403611834824762936835456, 2.44485299346754833530220528690, 2.96907015561519263112131630385, 3.02626542695282840096341419691, 3.09065626108276218281096289573, 3.42884314800694963558913232865, 3.70178779394171200360642346684, 3.72336615249092472991978150838, 3.80706272808054249979105609226, 3.87319799182977153118946223094, 3.92046411490126000369589655410, 4.40054040388122085218212097179, 4.55762298474706677696697097271, 4.57489138718291226465089263430, 4.62749405665831549897715318354, 5.02567788839632063552240878289

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