Properties

Label 10-2009e5-1.1-c1e5-0-0
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-s − 4·3-s − 3·4-s + 5·5-s + 4·6-s + 3·8-s + 9-s − 5·10-s + 2·11-s + 12·12-s − 5·13-s − 20·15-s + 3·16-s − 13·17-s − 18-s − 15·20-s − 2·22-s + 2·23-s − 12·24-s + 11·25-s + 5·26-s + 18·27-s − 5·29-s + 20·30-s − 17·31-s − 4·32-s − 8·33-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s − 3/2·4-s + 2.23·5-s + 1.63·6-s + 1.06·8-s + 1/3·9-s − 1.58·10-s + 0.603·11-s + 3.46·12-s − 1.38·13-s − 5.16·15-s + 3/4·16-s − 3.15·17-s − 0.235·18-s − 3.35·20-s − 0.426·22-s + 0.417·23-s − 2.44·24-s + 11/5·25-s + 0.980·26-s + 3.46·27-s − 0.928·29-s + 3.65·30-s − 3.05·31-s − 0.707·32-s − 1.39·33-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: induced by $\chi_{2009} (1, \cdot )$
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$C_2 \wr S_5$ \( 1 + T + p^{2} T^{2} + p^{2} T^{3} + 5 p T^{4} + 11 T^{5} + 5 p^{2} T^{6} + p^{4} T^{7} + p^{5} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 4 T + 5 p T^{2} + 38 T^{3} + 29 p T^{4} + 157 T^{5} + 29 p^{2} T^{6} + 38 p^{2} T^{7} + 5 p^{4} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 - p T + 14 T^{2} - 14 T^{3} - 11 T^{4} + 86 T^{5} - 11 p T^{6} - 14 p^{2} T^{7} + 14 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \)
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

−5.88501350241085958655476388276, −5.70658465295966520191905091239, −5.62593883081304205267153379519, −5.57186786180215759854275337691, −5.40908198310232626954349656470, −4.95342665838951298116840952704, −4.95149522931480885821527814426, −4.77608002990278672198734818311, −4.66540704997428635101415977902, −4.49966951751074924821877384455, −4.34545966809654338741183507500, −3.97907488030448197110278937260, −3.90201972871781149247036894347, −3.84654611908581700753496929413, −3.17200756969222704890183264599, −3.10596952704629512806139718781, −2.95092082041346917091136601894, −2.69480238715705784688559381245, −2.58138860830674377135684017116, −2.26178894877230404965869654861, −1.94216982177035905095460641142, −1.69192395454443625274249368119, −1.61818608331242024158537996988, −1.40197895365906947914838220522, −1.06648634595800888515613576231, 0, 0, 0, 0, 0, 1.06648634595800888515613576231, 1.40197895365906947914838220522, 1.61818608331242024158537996988, 1.69192395454443625274249368119, 1.94216982177035905095460641142, 2.26178894877230404965869654861, 2.58138860830674377135684017116, 2.69480238715705784688559381245, 2.95092082041346917091136601894, 3.10596952704629512806139718781, 3.17200756969222704890183264599, 3.84654611908581700753496929413, 3.90201972871781149247036894347, 3.97907488030448197110278937260, 4.34545966809654338741183507500, 4.49966951751074924821877384455, 4.66540704997428635101415977902, 4.77608002990278672198734818311, 4.95149522931480885821527814426, 4.95342665838951298116840952704, 5.40908198310232626954349656470, 5.57186786180215759854275337691, 5.62593883081304205267153379519, 5.70658465295966520191905091239, 5.88501350241085958655476388276

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