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Properties

Label 10-7440e5-1.1-c1e5-0-2
Degree $10$
Conductor $2.280\times 10^{19}$
Sign $-1$
Analytic cond. $7.40031\times 10^{8}$
Root an. cond. $7.70770$
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  − 5·3-s − 5·5-s − 3·7-s + 15·9-s − 5·11-s + 2·13-s + 25·15-s + 6·17-s − 9·19-s + 15·21-s + 3·23-s + 15·25-s − 35·27-s + 2·29-s − 5·31-s + 25·33-s + 15·35-s + 4·37-s − 10·39-s + 8·41-s − 7·43-s − 75·45-s + 2·47-s − 6·49-s − 30·51-s + 5·53-s + 25·55-s + ⋯
L(s)  = 1  − 2.88·3-s − 2.23·5-s − 1.13·7-s + 5·9-s − 1.50·11-s + 0.554·13-s + 6.45·15-s + 1.45·17-s − 2.06·19-s + 3.27·21-s + 0.625·23-s + 3·25-s − 6.73·27-s + 0.371·29-s − 0.898·31-s + 4.35·33-s + 2.53·35-s + 0.657·37-s − 1.60·39-s + 1.24·41-s − 1.06·43-s − 11.1·45-s + 0.291·47-s − 6/7·49-s − 4.20·51-s + 0.686·53-s + 3.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{20} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5}\)
Sign: $-1$
Analytic conductor: \(7.40031\times 10^{8}\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{20} \cdot 3^{5} \cdot 5^{5} \cdot 31^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{5} \)
5$C_1$ \( ( 1 + T )^{5} \)
31$C_1$ \( ( 1 + T )^{5} \)
good7$C_2 \wr S_5$ \( 1 + 3 T + 15 T^{2} + 4 p T^{3} + 170 T^{4} + 354 T^{5} + 170 p T^{6} + 4 p^{3} T^{7} + 15 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 5 T + 29 T^{2} + 148 T^{3} + 576 T^{4} + 1918 T^{5} + 576 p T^{6} + 148 p^{2} T^{7} + 29 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 - 2 T + 21 T^{2} + 44 T^{3} + 22 T^{4} + 1548 T^{5} + 22 p T^{6} + 44 p^{2} T^{7} + 21 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 6 T + 57 T^{2} - 240 T^{3} + 1558 T^{4} - 5204 T^{5} + 1558 p T^{6} - 240 p^{2} T^{7} + 57 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 9 T + 91 T^{2} + 540 T^{3} + 3350 T^{4} + 14534 T^{5} + 3350 p T^{6} + 540 p^{2} T^{7} + 91 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 3 T + 57 T^{2} - 196 T^{3} + 1768 T^{4} - 5554 T^{5} + 1768 p T^{6} - 196 p^{2} T^{7} + 57 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 - 2 T + 69 T^{2} - 124 T^{3} + 2838 T^{4} - 5876 T^{5} + 2838 p T^{6} - 124 p^{2} T^{7} + 69 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 4 T + 117 T^{2} - 356 T^{3} + 6086 T^{4} - 15520 T^{5} + 6086 p T^{6} - 356 p^{2} T^{7} + 117 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 8 T + 133 T^{2} - 704 T^{3} + 7074 T^{4} - 31344 T^{5} + 7074 p T^{6} - 704 p^{2} T^{7} + 133 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 7 T + 155 T^{2} + 804 T^{3} + 11550 T^{4} + 48506 T^{5} + 11550 p T^{6} + 804 p^{2} T^{7} + 155 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 2 T + 203 T^{2} - 312 T^{3} + 17674 T^{4} - 20620 T^{5} + 17674 p T^{6} - 312 p^{2} T^{7} + 203 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 5 T + 73 T^{2} - 524 T^{3} + 5722 T^{4} - 14638 T^{5} + 5722 p T^{6} - 524 p^{2} T^{7} + 73 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 6 T + 271 T^{2} + 1228 T^{3} + 30330 T^{4} + 103196 T^{5} + 30330 p T^{6} + 1228 p^{2} T^{7} + 271 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 16 T + 361 T^{2} - 3808 T^{3} + 46898 T^{4} - 345248 T^{5} + 46898 p T^{6} - 3808 p^{2} T^{7} + 361 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 22 T + 211 T^{2} + 1084 T^{3} + 6934 T^{4} + 60924 T^{5} + 6934 p T^{6} + 1084 p^{2} T^{7} + 211 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 9 T + 159 T^{2} - 1272 T^{3} + 15634 T^{4} - 108862 T^{5} + 15634 p T^{6} - 1272 p^{2} T^{7} + 159 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 - 9 T + 275 T^{2} - 2584 T^{3} + 34848 T^{4} - 279390 T^{5} + 34848 p T^{6} - 2584 p^{2} T^{7} + 275 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 15 T + 175 T^{2} + 1372 T^{3} + 12430 T^{4} + 76202 T^{5} + 12430 p T^{6} + 1372 p^{2} T^{7} + 175 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 8 T + 279 T^{2} - 2064 T^{3} + 36050 T^{4} - 236496 T^{5} + 36050 p T^{6} - 2064 p^{2} T^{7} + 279 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 17 T + 365 T^{2} - 3384 T^{3} + 45454 T^{4} - 325222 T^{5} + 45454 p T^{6} - 3384 p^{2} T^{7} + 365 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 18 T + 461 T^{2} - 6328 T^{3} + 87186 T^{4} - 890220 T^{5} + 87186 p T^{6} - 6328 p^{2} T^{7} + 461 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} \)
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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

−4.96958340997735078920689435625, −4.91359837272611779920442929246, −4.72573724368274635489877631737, −4.58459501513714498465310298041, −4.55461185441213861178480218483, −4.18744756572320911645957085918, −4.15429038412858646290606787680, −4.06114194816075639556478356793, −3.79851388131795883158804712300, −3.70677294095359529843873470308, −3.53102004074312726464505003427, −3.30624608501298123819685132627, −3.24868793261115651049577020972, −3.12057949248167692643765937104, −2.98738463029968420116607781915, −2.41994508191683271554164130638, −2.39505879620166464064007809387, −2.31715711060155171867312121610, −2.12777903130393547231065417920, −2.00943571669209962936284411196, −1.31142455090167143081737890396, −1.14041693243456340293567806728, −1.13606657132198305341019434090, −0.973184252727378876839935444537, −0.966124232233696994084201142579, 0, 0, 0, 0, 0, 0.966124232233696994084201142579, 0.973184252727378876839935444537, 1.13606657132198305341019434090, 1.14041693243456340293567806728, 1.31142455090167143081737890396, 2.00943571669209962936284411196, 2.12777903130393547231065417920, 2.31715711060155171867312121610, 2.39505879620166464064007809387, 2.41994508191683271554164130638, 2.98738463029968420116607781915, 3.12057949248167692643765937104, 3.24868793261115651049577020972, 3.30624608501298123819685132627, 3.53102004074312726464505003427, 3.70677294095359529843873470308, 3.79851388131795883158804712300, 4.06114194816075639556478356793, 4.15429038412858646290606787680, 4.18744756572320911645957085918, 4.55461185441213861178480218483, 4.58459501513714498465310298041, 4.72573724368274635489877631737, 4.91359837272611779920442929246, 4.96958340997735078920689435625

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