Properties

Label 10-7175e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.902\times 10^{19}$
Sign $-1$
Analytic cond. $6.17298\times 10^{8}$
Root an. cond. $7.56919$
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  + 2-s − 4·3-s − 3·4-s − 4·6-s − 5·7-s − 3·8-s + 9-s + 2·11-s + 12·12-s − 5·13-s − 5·14-s + 3·16-s − 13·17-s + 18-s + 20·21-s + 2·22-s − 2·23-s + 12·24-s − 5·26-s + 18·27-s + 15·28-s − 5·29-s + 17·31-s + 4·32-s − 8·33-s − 13·34-s − 3·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s − 3/2·4-s − 1.63·6-s − 1.88·7-s − 1.06·8-s + 1/3·9-s + 0.603·11-s + 3.46·12-s − 1.38·13-s − 1.33·14-s + 3/4·16-s − 3.15·17-s + 0.235·18-s + 4.36·21-s + 0.426·22-s − 0.417·23-s + 2.44·24-s − 0.980·26-s + 3.46·27-s + 2.83·28-s − 0.928·29-s + 3.05·31-s + 0.707·32-s − 1.39·33-s − 2.22·34-s − 1/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{5} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 7^{5} \cdot 41^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{10} \cdot 7^{5} \cdot 41^{5}\)
Sign: $-1$
Analytic conductor: \(6.17298\times 10^{8}\)
Root analytic conductor: \(7.56919\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 5^{10} \cdot 7^{5} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad5 \( 1 \)
7$C_1$ \( ( 1 + T )^{5} \)
41$C_1$ \( ( 1 + T )^{5} \)
good2$C_2 \wr S_5$ \( 1 - T + p^{2} T^{2} - p^{2} T^{3} + 5 p T^{4} - 11 T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{5} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 4 T + 5 p T^{2} + 38 T^{3} + 29 p T^{4} + 157 T^{5} + 29 p^{2} T^{6} + 38 p^{2} T^{7} + 5 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 2 T - 8 T^{2} + 52 T^{3} + 103 T^{4} - 844 T^{5} + 103 p T^{6} + 52 p^{2} T^{7} - 8 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 5 T + 56 T^{2} + 180 T^{3} + 1219 T^{4} + 2941 T^{5} + 1219 p T^{6} + 180 p^{2} T^{7} + 56 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 13 T + 110 T^{2} + 580 T^{3} + 2611 T^{4} + 10157 T^{5} + 2611 p T^{6} + 580 p^{2} T^{7} + 110 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 47 T^{2} - 132 T^{3} + 851 T^{4} - 5017 T^{5} + 851 p T^{6} - 132 p^{2} T^{7} + 47 p^{3} T^{8} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 2 T + 49 T^{2} + 158 T^{3} + 1885 T^{4} + 3835 T^{5} + 1885 p T^{6} + 158 p^{2} T^{7} + 49 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 5 T + 74 T^{2} + 230 T^{3} + 2629 T^{4} + 6442 T^{5} + 2629 p T^{6} + 230 p^{2} T^{7} + 74 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 17 T + 226 T^{2} - 2084 T^{3} + 16065 T^{4} - 96590 T^{5} + 16065 p T^{6} - 2084 p^{2} T^{7} + 226 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 7 T + 149 T^{2} - 879 T^{3} + 9772 T^{4} - 45884 T^{5} + 9772 p T^{6} - 879 p^{2} T^{7} + 149 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + T + 102 T^{2} + 198 T^{3} + 6135 T^{4} + 15081 T^{5} + 6135 p T^{6} + 198 p^{2} T^{7} + 102 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 9 T + 115 T^{2} + 653 T^{3} + 5488 T^{4} + 31712 T^{5} + 5488 p T^{6} + 653 p^{2} T^{7} + 115 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 5 T + 156 T^{2} + 36 T^{3} + 7939 T^{4} - 26602 T^{5} + 7939 p T^{6} + 36 p^{2} T^{7} + 156 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 7 T + 124 T^{2} - 772 T^{3} + 12043 T^{4} - 63362 T^{5} + 12043 p T^{6} - 772 p^{2} T^{7} + 124 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 22 T + 458 T^{2} - 5620 T^{3} + 64261 T^{4} - 519412 T^{5} + 64261 p T^{6} - 5620 p^{2} T^{7} + 458 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 3 T + 230 T^{2} - 228 T^{3} + 22853 T^{4} - 3146 T^{5} + 22853 p T^{6} - 228 p^{2} T^{7} + 230 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 24 T + 396 T^{2} + 4954 T^{3} + 55267 T^{4} + 504628 T^{5} + 55267 p T^{6} + 4954 p^{2} T^{7} + 396 p^{3} T^{8} + 24 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 40 T + 950 T^{2} + 15562 T^{3} + 192937 T^{4} + 1857916 T^{5} + 192937 p T^{6} + 15562 p^{2} T^{7} + 950 p^{3} T^{8} + 40 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 42 T + 967 T^{2} + 15192 T^{3} + 184214 T^{4} + 1801084 T^{5} + 184214 p T^{6} + 15192 p^{2} T^{7} + 967 p^{3} T^{8} + 42 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 12 T + 156 T^{2} - 1442 T^{3} + 16075 T^{4} - 98732 T^{5} + 16075 p T^{6} - 1442 p^{2} T^{7} + 156 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 8 T + 457 T^{2} - 2846 T^{3} + 82405 T^{4} - 379849 T^{5} + 82405 p T^{6} - 2846 p^{2} T^{7} + 457 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 16 T + 323 T^{2} + 4394 T^{3} + 51905 T^{4} + 561841 T^{5} + 51905 p T^{6} + 4394 p^{2} T^{7} + 323 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.96495402010379907790172574620, −4.87305491906945793281620210990, −4.73297583498016428898312020533, −4.64245345141319251301791566611, −4.50780101931644657880012328025, −4.28134095724044682509769737833, −4.15413486511398367093031771154, −4.12944862302526178278440983435, −4.05126829041983711163576076456, −3.83565761714192345552058566840, −3.33416820533995645221540356043, −3.32220781228903562843790656921, −3.26272753866645483608738162656, −3.04857008400630583736892980286, −2.76893399825455681641396379830, −2.66097505671485450455245365099, −2.56324986459349791463098566261, −2.41149583932992934612256938088, −2.16379314520324087983234846178, −1.89046902478350222290913944487, −1.80194792517392454727827047202, −1.29710268924653848517522682015, −1.06310523626560540330920091818, −0.944318240423707446270920267077, −0.66357386477421182045039087437, 0, 0, 0, 0, 0, 0.66357386477421182045039087437, 0.944318240423707446270920267077, 1.06310523626560540330920091818, 1.29710268924653848517522682015, 1.80194792517392454727827047202, 1.89046902478350222290913944487, 2.16379314520324087983234846178, 2.41149583932992934612256938088, 2.56324986459349791463098566261, 2.66097505671485450455245365099, 2.76893399825455681641396379830, 3.04857008400630583736892980286, 3.26272753866645483608738162656, 3.32220781228903562843790656921, 3.33416820533995645221540356043, 3.83565761714192345552058566840, 4.05126829041983711163576076456, 4.12944862302526178278440983435, 4.15413486511398367093031771154, 4.28134095724044682509769737833, 4.50780101931644657880012328025, 4.64245345141319251301791566611, 4.73297583498016428898312020533, 4.87305491906945793281620210990, 4.96495402010379907790172574620

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