Properties

Label 2-10-1.1-c11-0-4
Degree $2$
Conductor $10$
Sign $-1$
Analytic cond. $7.68343$
Root an. cond. $2.77190$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s − 318·3-s + 1.02e3·4-s − 3.12e3·5-s − 1.01e4·6-s − 7.07e4·7-s + 3.27e4·8-s − 7.60e4·9-s − 1.00e5·10-s + 2.38e5·11-s − 3.25e5·12-s − 2.09e6·13-s − 2.26e6·14-s + 9.93e5·15-s + 1.04e6·16-s + 5.95e6·17-s − 2.43e6·18-s + 1.02e7·19-s − 3.20e6·20-s + 2.24e7·21-s + 7.62e6·22-s − 3.53e6·23-s − 1.04e7·24-s + 9.76e6·25-s − 6.71e7·26-s + 8.05e7·27-s − 7.24e7·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.755·3-s + 1/2·4-s − 0.447·5-s − 0.534·6-s − 1.59·7-s + 0.353·8-s − 0.429·9-s − 0.316·10-s + 0.446·11-s − 0.377·12-s − 1.56·13-s − 1.12·14-s + 0.337·15-s + 1/4·16-s + 1.01·17-s − 0.303·18-s + 0.946·19-s − 0.223·20-s + 1.20·21-s + 0.315·22-s − 0.114·23-s − 0.267·24-s + 1/5·25-s − 1.10·26-s + 1.07·27-s − 0.795·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-1$
Analytic conductor: \(7.68343\)
Root analytic conductor: \(2.77190\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{5} T \)
5 \( 1 + p^{5} T \)
good3 \( 1 + 106 p T + p^{11} T^{2} \)
7 \( 1 + 10102 p T + p^{11} T^{2} \)
11 \( 1 - 238272 T + p^{11} T^{2} \)
13 \( 1 + 2097478 T + p^{11} T^{2} \)
17 \( 1 - 5955546 T + p^{11} T^{2} \)
19 \( 1 - 10210820 T + p^{11} T^{2} \)
23 \( 1 + 3535758 T + p^{11} T^{2} \)
29 \( 1 + 139304850 T + p^{11} T^{2} \)
31 \( 1 + 101002348 T + p^{11} T^{2} \)
37 \( 1 + 524913814 T + p^{11} T^{2} \)
41 \( 1 - 284590422 T + p^{11} T^{2} \)
43 \( 1 + 1253635078 T + p^{11} T^{2} \)
47 \( 1 + 216106434 T + p^{11} T^{2} \)
53 \( 1 + 4881275358 T + p^{11} T^{2} \)
59 \( 1 - 8692473300 T + p^{11} T^{2} \)
61 \( 1 - 3296491802 T + p^{11} T^{2} \)
67 \( 1 - 18275027966 T + p^{11} T^{2} \)
71 \( 1 + 13287447588 T + p^{11} T^{2} \)
73 \( 1 + 32505250798 T + p^{11} T^{2} \)
79 \( 1 - 9297455960 T + p^{11} T^{2} \)
83 \( 1 + 22741484838 T + p^{11} T^{2} \)
89 \( 1 + 93378882390 T + p^{11} T^{2} \)
97 \( 1 + 5811134014 T + p^{11} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.01618597135081896713273389127, −16.11601892396703700439948601753, −14.49506184906278364109210905204, −12.67429178840850196346152466421, −11.72100741358271812309889062364, −9.829838726570310491095976970921, −7.04509828040427304497400589835, −5.47434125415943491018676469462, −3.28219328272675841331927180697, 0, 3.28219328272675841331927180697, 5.47434125415943491018676469462, 7.04509828040427304497400589835, 9.829838726570310491095976970921, 11.72100741358271812309889062364, 12.67429178840850196346152466421, 14.49506184906278364109210905204, 16.11601892396703700439948601753, 17.01618597135081896713273389127

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