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Properties

Label 10-7400e5-1.1-c1e5-0-1
Degree $10$
Conductor $2.219\times 10^{19}$
Sign $1$
Analytic cond. $7.20350\times 10^{8}$
Root an. cond. $7.68695$
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-s + 2·7-s − 4·9-s + 8·11-s + 4·13-s + 17-s + 10·19-s − 2·21-s − 4·23-s + 5·27-s + 11·29-s − 3·31-s − 8·33-s + 5·37-s − 4·39-s + 16·41-s − 31·43-s − 5·47-s − 12·49-s − 51-s − 8·53-s − 10·57-s + 24·59-s − 15·61-s − 8·63-s − 12·67-s + 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s − 4/3·9-s + 2.41·11-s + 1.10·13-s + 0.242·17-s + 2.29·19-s − 0.436·21-s − 0.834·23-s + 0.962·27-s + 2.04·29-s − 0.538·31-s − 1.39·33-s + 0.821·37-s − 0.640·39-s + 2.49·41-s − 4.72·43-s − 0.729·47-s − 1.71·49-s − 0.140·51-s − 1.09·53-s − 1.32·57-s + 3.12·59-s − 1.92·61-s − 1.00·63-s − 1.46·67-s + 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{10} \cdot 37^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{15} \cdot 5^{10} \cdot 37^{5}\)
Sign: $1$
Analytic conductor: \(7.20350\times 10^{8}\)
Root analytic conductor: \(7.68695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((10,\ 2^{15} \cdot 5^{10} \cdot 37^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.363341796\)
\(L(\frac12)\) \(\approx\) \(5.363341796\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
37$C_1$ \( ( 1 - T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + T + 5 T^{2} + 4 T^{3} + 4 p T^{4} + 14 T^{5} + 4 p^{2} T^{6} + 4 p^{2} T^{7} + 5 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 - 2 T + 16 T^{2} - 40 T^{3} + 191 T^{4} - 312 T^{5} + 191 p T^{6} - 40 p^{2} T^{7} + 16 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 8 T + 6 p T^{2} - 324 T^{3} + 1541 T^{4} - 5224 T^{5} + 1541 p T^{6} - 324 p^{2} T^{7} + 6 p^{4} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 4 T + 41 T^{2} - 128 T^{3} + 802 T^{4} - 2040 T^{5} + 802 p T^{6} - 128 p^{2} T^{7} + 41 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - T + 35 T^{2} - 48 T^{3} + 636 T^{4} - 926 T^{5} + 636 p T^{6} - 48 p^{2} T^{7} + 35 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 10 T + 99 T^{2} - 612 T^{3} + 3590 T^{4} - 16068 T^{5} + 3590 p T^{6} - 612 p^{2} T^{7} + 99 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 4 T + 63 T^{2} + 224 T^{3} + 2278 T^{4} + 6328 T^{5} + 2278 p T^{6} + 224 p^{2} T^{7} + 63 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 11 T + 117 T^{2} - 860 T^{3} + 5686 T^{4} - 31586 T^{5} + 5686 p T^{6} - 860 p^{2} T^{7} + 117 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 3 T + 59 T^{2} + 156 T^{3} + 2622 T^{4} + 8642 T^{5} + 2622 p T^{6} + 156 p^{2} T^{7} + 59 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 16 T + 162 T^{2} - 1190 T^{3} + 9357 T^{4} - 62804 T^{5} + 9357 p T^{6} - 1190 p^{2} T^{7} + 162 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 31 T + 509 T^{2} + 5716 T^{3} + 49760 T^{4} + 355946 T^{5} + 49760 p T^{6} + 5716 p^{2} T^{7} + 509 p^{3} T^{8} + 31 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 5 T + 71 T^{2} + 220 T^{3} + 4442 T^{4} + 23534 T^{5} + 4442 p T^{6} + 220 p^{2} T^{7} + 71 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 8 T + 186 T^{2} + 1054 T^{3} + 16293 T^{4} + 74884 T^{5} + 16293 p T^{6} + 1054 p^{2} T^{7} + 186 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 24 T + 451 T^{2} - 6036 T^{3} + 62790 T^{4} - 545288 T^{5} + 62790 p T^{6} - 6036 p^{2} T^{7} + 451 p^{3} T^{8} - 24 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 15 T + 159 T^{2} + 2160 T^{3} + 19988 T^{4} + 139282 T^{5} + 19988 p T^{6} + 2160 p^{2} T^{7} + 159 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 12 T + 147 T^{2} + 12 T^{3} - 6898 T^{4} - 125200 T^{5} - 6898 p T^{6} + 12 p^{2} T^{7} + 147 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 5 T + 277 T^{2} - 1308 T^{3} + 35204 T^{4} - 134302 T^{5} + 35204 p T^{6} - 1308 p^{2} T^{7} + 277 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 5 T + 287 T^{2} + 1064 T^{3} + 37048 T^{4} + 105670 T^{5} + 37048 p T^{6} + 1064 p^{2} T^{7} + 287 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 4 T + 223 T^{2} - 1124 T^{3} + 22318 T^{4} - 128176 T^{5} + 22318 p T^{6} - 1124 p^{2} T^{7} + 223 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 27 T + 527 T^{2} + 6556 T^{3} + 72526 T^{4} + 655314 T^{5} + 72526 p T^{6} + 6556 p^{2} T^{7} + 527 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 16 T + 317 T^{2} - 2896 T^{3} + 34778 T^{4} - 252480 T^{5} + 34778 p T^{6} - 2896 p^{2} T^{7} + 317 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 19 T + 539 T^{2} - 7208 T^{3} + 109668 T^{4} - 1040842 T^{5} + 109668 p T^{6} - 7208 p^{2} T^{7} + 539 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.60942855975033205062224305207, −4.48511847704335746762009582219, −4.35098588894009955510650923588, −4.25182454130909138056614689271, −3.94882351230333466773824194711, −3.88747780488089588959557151587, −3.67490978600623033771106043638, −3.51062946284997318547497724090, −3.37913074431920889327704371282, −3.29277903833306657787849659757, −3.13904721609893799393389524952, −2.78480738848894428353271657655, −2.70459107919047340779852758266, −2.67792861700451371800596957649, −2.59120528102726719017043864235, −1.92634728746472502398629219525, −1.82015595739351731900856462289, −1.65973466355311946288202145383, −1.63536207338878523633743513199, −1.41940501825076066721651336512, −1.22787994787611650738828539737, −0.936621535763951946928421126142, −0.75336495096653687808974968719, −0.53360558606245300433544975569, −0.22052923243683105048120332584, 0.22052923243683105048120332584, 0.53360558606245300433544975569, 0.75336495096653687808974968719, 0.936621535763951946928421126142, 1.22787994787611650738828539737, 1.41940501825076066721651336512, 1.63536207338878523633743513199, 1.65973466355311946288202145383, 1.82015595739351731900856462289, 1.92634728746472502398629219525, 2.59120528102726719017043864235, 2.67792861700451371800596957649, 2.70459107919047340779852758266, 2.78480738848894428353271657655, 3.13904721609893799393389524952, 3.29277903833306657787849659757, 3.37913074431920889327704371282, 3.51062946284997318547497724090, 3.67490978600623033771106043638, 3.88747780488089588959557151587, 3.94882351230333466773824194711, 4.25182454130909138056614689271, 4.35098588894009955510650923588, 4.48511847704335746762009582219, 4.60942855975033205062224305207

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