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Properties

Label 10-2009e5-1.1-c1e5-0-1
Degree $10$
Conductor $3.273\times 10^{16}$
Sign $-1$
Analytic cond. $1.06239\times 10^{6}$
Root an. cond. $4.00523$
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  − 2·2-s + 2·3-s − 2·5-s − 4·6-s + 3·8-s − 3·9-s + 4·10-s − 6·11-s − 2·13-s − 4·15-s − 8·16-s + 3·17-s + 6·18-s − 7·19-s + 12·22-s + 6·24-s − 13·25-s + 4·26-s − 15·27-s − 10·29-s + 8·30-s + 6·31-s + 11·32-s − 12·33-s − 6·34-s − 18·37-s + 14·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s − 0.894·5-s − 1.63·6-s + 1.06·8-s − 9-s + 1.26·10-s − 1.80·11-s − 0.554·13-s − 1.03·15-s − 2·16-s + 0.727·17-s + 1.41·18-s − 1.60·19-s + 2.55·22-s + 1.22·24-s − 2.59·25-s + 0.784·26-s − 2.88·27-s − 1.85·29-s + 1.46·30-s + 1.07·31-s + 1.94·32-s − 2.08·33-s − 1.02·34-s − 2.95·37-s + 2.27·38-s + ⋯

Functional equation

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

Invariants

Degree: \(10\)
Conductor: \(7^{10} \cdot 41^{5}\)
Sign: $-1$
Analytic conductor: \(1.06239\times 10^{6}\)
Root analytic conductor: \(4.00523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 7^{10} \cdot 41^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad7 \( 1 \)
41$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + p^{2} T^{2} + 5 T^{3} + 3 p^{2} T^{4} + 17 T^{5} + 3 p^{3} T^{6} + 5 p^{2} T^{7} + p^{5} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 - 2 T + 7 T^{2} - 5 T^{3} + 16 T^{4} - T^{5} + 16 p T^{6} - 5 p^{2} T^{7} + 7 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 2 T + 17 T^{2} + 21 T^{3} + 128 T^{4} + 117 T^{5} + 128 p T^{6} + 21 p^{2} T^{7} + 17 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 6 T + 52 T^{2} + 219 T^{3} + 1118 T^{4} + 3419 T^{5} + 1118 p T^{6} + 219 p^{2} T^{7} + 52 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 2 T + 57 T^{2} + 85 T^{3} + 1376 T^{4} + 1541 T^{5} + 1376 p T^{6} + 85 p^{2} T^{7} + 57 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 3 T + 60 T^{2} - 9 p T^{3} + 1723 T^{4} - 3629 T^{5} + 1723 p T^{6} - 9 p^{3} T^{7} + 60 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 7 T + 79 T^{2} + 391 T^{3} + 2739 T^{4} + 10441 T^{5} + 2739 p T^{6} + 391 p^{2} T^{7} + 79 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 77 T^{2} + 19 T^{3} + 2870 T^{4} + 683 T^{5} + 2870 p T^{6} + 19 p^{2} T^{7} + 77 p^{3} T^{8} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 10 T + 154 T^{2} + 1024 T^{3} + 8899 T^{4} + 42621 T^{5} + 8899 p T^{6} + 1024 p^{2} T^{7} + 154 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 6 T + 3 p T^{2} - 515 T^{3} + 4802 T^{4} - 20097 T^{5} + 4802 p T^{6} - 515 p^{2} T^{7} + 3 p^{4} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 18 T + 250 T^{2} + 66 p T^{3} + 20191 T^{4} + 131177 T^{5} + 20191 p T^{6} + 66 p^{3} T^{7} + 250 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 14 T + 162 T^{2} + 1230 T^{3} + 7725 T^{4} + 51435 T^{5} + 7725 p T^{6} + 1230 p^{2} T^{7} + 162 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 3 T + 148 T^{2} + 509 T^{3} + 10681 T^{4} + 35789 T^{5} + 10681 p T^{6} + 509 p^{2} T^{7} + 148 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 9 T + 145 T^{2} + 535 T^{3} + 6097 T^{4} + 4521 T^{5} + 6097 p T^{6} + 535 p^{2} T^{7} + 145 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 19 T + 292 T^{2} - 3271 T^{3} + 33435 T^{4} - 271711 T^{5} + 33435 p T^{6} - 3271 p^{2} T^{7} + 292 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 23 T + 321 T^{2} - 4061 T^{3} + 41031 T^{4} - 336925 T^{5} + 41031 p T^{6} - 4061 p^{2} T^{7} + 321 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 11 T + 200 T^{2} + 1605 T^{3} + 20103 T^{4} + 135609 T^{5} + 20103 p T^{6} + 1605 p^{2} T^{7} + 200 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 205 T^{2} + 61 T^{3} + 23652 T^{4} + 1421 T^{5} + 23652 p T^{6} + 61 p^{2} T^{7} + 205 p^{3} T^{8} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 13 T + 255 T^{2} + 2110 T^{3} + 24189 T^{4} + 166475 T^{5} + 24189 p T^{6} + 2110 p^{2} T^{7} + 255 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 41 T + 12 p T^{2} + 15081 T^{3} + 184951 T^{4} + 1815285 T^{5} + 184951 p T^{6} + 15081 p^{2} T^{7} + 12 p^{4} T^{8} + 41 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 2 T + 338 T^{2} - 520 T^{3} + 50961 T^{4} - 61081 T^{5} + 50961 p T^{6} - 520 p^{2} T^{7} + 338 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 14 T + 194 T^{2} + 1342 T^{3} + 12493 T^{4} + 62707 T^{5} + 12493 p T^{6} + 1342 p^{2} T^{7} + 194 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 27 T + 448 T^{2} - 5083 T^{3} + 45427 T^{4} - 421253 T^{5} + 45427 p T^{6} - 5083 p^{2} T^{7} + 448 p^{3} T^{8} - 27 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.87028083685713191162206723184, −5.63814265801489371540924025559, −5.56582269840245890241127360623, −5.48652888055762692082800915744, −5.26451577702681737435381246733, −5.06933145586121281558283843865, −4.91518548086244915468771200474, −4.63567764919776045539140174906, −4.45339515685719469401413940419, −4.22275042519382888456620944709, −4.13719106430963042701490959856, −3.76623688584059759615992868161, −3.75231396275051602368530310248, −3.72893893017871731861718286643, −3.35164832487139713786886452159, −3.09171513384321421615814405059, −3.03408454942780318134378070389, −2.57501438447230924694328129706, −2.51972438404041901459294596558, −2.27436390961543582730374358753, −2.17459746040910271279346551381, −2.08756824820153473739133608155, −1.53902251922841008397716756830, −1.36529399632823505451369982604, −1.20625702396805297936603425263, 0, 0, 0, 0, 0, 1.20625702396805297936603425263, 1.36529399632823505451369982604, 1.53902251922841008397716756830, 2.08756824820153473739133608155, 2.17459746040910271279346551381, 2.27436390961543582730374358753, 2.51972438404041901459294596558, 2.57501438447230924694328129706, 3.03408454942780318134378070389, 3.09171513384321421615814405059, 3.35164832487139713786886452159, 3.72893893017871731861718286643, 3.75231396275051602368530310248, 3.76623688584059759615992868161, 4.13719106430963042701490959856, 4.22275042519382888456620944709, 4.45339515685719469401413940419, 4.63567764919776045539140174906, 4.91518548086244915468771200474, 5.06933145586121281558283843865, 5.26451577702681737435381246733, 5.48652888055762692082800915744, 5.56582269840245890241127360623, 5.63814265801489371540924025559, 5.87028083685713191162206723184

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