↑

Properties

Label 10-7605e5-1.1-c1e5-0-2
Degree $10$
Conductor $2.544\times 10^{19}$
Sign $-1$
Analytic cond. $8.25812\times 10^{8}$
Root an. cond. $7.79270$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 5·5-s − 7-s + 2·8-s + 10·10-s − 8·11-s + 2·14-s − 2·16-s + 4·19-s + 16·22-s − 6·23-s + 15·25-s + 16·29-s − 9·31-s + 2·32-s + 5·35-s + 4·37-s − 8·38-s − 10·40-s − 6·41-s + 15·43-s + 12·46-s − 10·47-s − 12·49-s − 30·50-s − 20·53-s + 40·55-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.23·5-s − 0.377·7-s + 0.707·8-s + 3.16·10-s − 2.41·11-s + 0.534·14-s − 1/2·16-s + 0.917·19-s + 3.41·22-s − 1.25·23-s + 3·25-s + 2.97·29-s − 1.61·31-s + 0.353·32-s + 0.845·35-s + 0.657·37-s − 1.29·38-s − 1.58·40-s − 0.937·41-s + 2.28·43-s + 1.76·46-s − 1.45·47-s − 1.71·49-s − 4.24·50-s − 2.74·53-s + 5.39·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{5} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{5} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(3^{10} \cdot 5^{5} \cdot 13^{10}\)
Sign: $-1$
Analytic conductor: \(8.25812\times 10^{8}\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 3^{10} \cdot 5^{5} \cdot 13^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad3 \( 1 \)
5$C_1$ \( ( 1 + T )^{5} \)
13 \( 1 \)
good2$C_2 \wr S_5$ \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + 5 p T^{4} + 7 p T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{5} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + T + 13 T^{2} + 6 T^{3} + 75 T^{4} + 5 p T^{5} + 75 p T^{6} + 6 p^{2} T^{7} + 13 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 8 T + 5 p T^{2} + 288 T^{3} + 1258 T^{4} + 4424 T^{5} + 1258 p T^{6} + 288 p^{2} T^{7} + 5 p^{4} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 35 T^{2} - 14 T^{3} + 910 T^{4} + 166 T^{5} + 910 p T^{6} - 14 p^{2} T^{7} + 35 p^{3} T^{8} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 4 T + 87 T^{2} - 268 T^{3} + 3150 T^{4} - 7308 T^{5} + 3150 p T^{6} - 268 p^{2} T^{7} + 87 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 6 T + 95 T^{2} + 516 T^{3} + 4030 T^{4} + 17316 T^{5} + 4030 p T^{6} + 516 p^{2} T^{7} + 95 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 16 T + 7 p T^{2} - 1764 T^{3} + 13072 T^{4} - 75142 T^{5} + 13072 p T^{6} - 1764 p^{2} T^{7} + 7 p^{4} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 9 T + 125 T^{2} + 938 T^{3} + 7513 T^{4} + 40255 T^{5} + 7513 p T^{6} + 938 p^{2} T^{7} + 125 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 121 T^{2} - 648 T^{3} + 6826 T^{4} - 36856 T^{5} + 6826 p T^{6} - 648 p^{2} T^{7} + 121 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 6 T + 107 T^{2} + 420 T^{3} + 6892 T^{4} + 26166 T^{5} + 6892 p T^{6} + 420 p^{2} T^{7} + 107 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 - 15 T + 245 T^{2} - 2122 T^{3} + 20133 T^{4} - 124963 T^{5} + 20133 p T^{6} - 2122 p^{2} T^{7} + 245 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 10 T + 145 T^{2} + 24 p T^{3} + 10018 T^{4} + 69286 T^{5} + 10018 p T^{6} + 24 p^{3} T^{7} + 145 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 20 T + 5 p T^{2} + 2368 T^{3} + 17722 T^{4} + 123096 T^{5} + 17722 p T^{6} + 2368 p^{2} T^{7} + 5 p^{4} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 12 T + 253 T^{2} + 2436 T^{3} + 27700 T^{4} + 206658 T^{5} + 27700 p T^{6} + 2436 p^{2} T^{7} + 253 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 11 T + 291 T^{2} - 2278 T^{3} + 34301 T^{4} - 199077 T^{5} + 34301 p T^{6} - 2278 p^{2} T^{7} + 291 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 5 T + 291 T^{2} + 1336 T^{3} + 36125 T^{4} + 134151 T^{5} + 36125 p T^{6} + 1336 p^{2} T^{7} + 291 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 10 T + 333 T^{2} + 2486 T^{3} + 45208 T^{4} + 252030 T^{5} + 45208 p T^{6} + 2486 p^{2} T^{7} + 333 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + T + 129 T^{2} + 148 T^{3} + 13711 T^{4} + 1257 T^{5} + 13711 p T^{6} + 148 p^{2} T^{7} + 129 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 17 T + 369 T^{2} + 4286 T^{3} + 54349 T^{4} + 467343 T^{5} + 54349 p T^{6} + 4286 p^{2} T^{7} + 369 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 16 T + 299 T^{2} + 3252 T^{3} + 38638 T^{4} + 325648 T^{5} + 38638 p T^{6} + 3252 p^{2} T^{7} + 299 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 4 T + 343 T^{2} + 1346 T^{3} + 52048 T^{4} + 176274 T^{5} + 52048 p T^{6} + 1346 p^{2} T^{7} + 343 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 11 T + 355 T^{2} - 2162 T^{3} + 48109 T^{4} - 204751 T^{5} + 48109 p T^{6} - 2162 p^{2} T^{7} + 355 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.80636018293257970272258977908, −4.75303036125191711463477067283, −4.69313736746484964595923327001, −4.67771988527906196230031227606, −4.64322147094008612359348219202, −4.09970338570279864790702742590, −4.00728070588478890790890605845, −3.95422976186580377088765206850, −3.80632891820890186812784761690, −3.60292345149754756855289065869, −3.39113545947333362020621572911, −3.36432700719367679150755225072, −3.02672953888810707765626627016, −2.90758855393346008119799880634, −2.74440409844030969965217157222, −2.72523740049674796101982416094, −2.47933752754595346655430390793, −2.41766016573606296738230935891, −2.10476810251785245405687119718, −1.70122059937139778289772822334, −1.52571648749590368048754386781, −1.32311288674600197273345476880, −1.24931277291513498918825331732, −0.890651908953595927553360581564, −0.76654662970760612084386123505, 0, 0, 0, 0, 0, 0.76654662970760612084386123505, 0.890651908953595927553360581564, 1.24931277291513498918825331732, 1.32311288674600197273345476880, 1.52571648749590368048754386781, 1.70122059937139778289772822334, 2.10476810251785245405687119718, 2.41766016573606296738230935891, 2.47933752754595346655430390793, 2.72523740049674796101982416094, 2.74440409844030969965217157222, 2.90758855393346008119799880634, 3.02672953888810707765626627016, 3.36432700719367679150755225072, 3.39113545947333362020621572911, 3.60292345149754756855289065869, 3.80632891820890186812784761690, 3.95422976186580377088765206850, 4.00728070588478890790890605845, 4.09970338570279864790702742590, 4.64322147094008612359348219202, 4.67771988527906196230031227606, 4.69313736746484964595923327001, 4.75303036125191711463477067283, 4.80636018293257970272258977908

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