Properties

Label 2-17e2-1.1-c3-0-45
Degree $2$
Conductor $289$
Sign $1$
Analytic cond. $17.0515$
Root an. cond. $4.12935$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.42·2-s + 9.44·3-s + 11.5·4-s − 8.63·5-s + 41.7·6-s + 12.6·7-s + 15.7·8-s + 62.2·9-s − 38.1·10-s + 14.5·11-s + 109.·12-s − 26.6·13-s + 55.9·14-s − 81.5·15-s − 22.7·16-s + 275.·18-s − 125.·19-s − 99.8·20-s + 119.·21-s + 64.2·22-s − 2.31·23-s + 149.·24-s − 50.5·25-s − 117.·26-s + 333.·27-s + 146.·28-s − 21.8·29-s + ⋯
L(s)  = 1  + 1.56·2-s + 1.81·3-s + 1.44·4-s − 0.771·5-s + 2.84·6-s + 0.683·7-s + 0.696·8-s + 2.30·9-s − 1.20·10-s + 0.397·11-s + 2.62·12-s − 0.568·13-s + 1.06·14-s − 1.40·15-s − 0.355·16-s + 3.60·18-s − 1.51·19-s − 1.11·20-s + 1.24·21-s + 0.622·22-s − 0.0210·23-s + 1.26·24-s − 0.404·25-s − 0.889·26-s + 2.37·27-s + 0.987·28-s − 0.139·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $1$
Analytic conductor: \(17.0515\)
Root analytic conductor: \(4.12935\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.016487658\)
\(L(\frac12)\) \(\approx\) \(7.016487658\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 - 4.42T + 8T^{2} \)
3 \( 1 - 9.44T + 27T^{2} \)
5 \( 1 + 8.63T + 125T^{2} \)
7 \( 1 - 12.6T + 343T^{2} \)
11 \( 1 - 14.5T + 1.33e3T^{2} \)
13 \( 1 + 26.6T + 2.19e3T^{2} \)
19 \( 1 + 125.T + 6.85e3T^{2} \)
23 \( 1 + 2.31T + 1.21e4T^{2} \)
29 \( 1 + 21.8T + 2.43e4T^{2} \)
31 \( 1 - 323.T + 2.97e4T^{2} \)
37 \( 1 - 73.2T + 5.06e4T^{2} \)
41 \( 1 + 179.T + 6.89e4T^{2} \)
43 \( 1 - 186.T + 7.95e4T^{2} \)
47 \( 1 - 235.T + 1.03e5T^{2} \)
53 \( 1 - 200.T + 1.48e5T^{2} \)
59 \( 1 + 718.T + 2.05e5T^{2} \)
61 \( 1 + 727.T + 2.26e5T^{2} \)
67 \( 1 - 76.9T + 3.00e5T^{2} \)
71 \( 1 + 923.T + 3.57e5T^{2} \)
73 \( 1 - 820.T + 3.89e5T^{2} \)
79 \( 1 - 51.0T + 4.93e5T^{2} \)
83 \( 1 - 1.18e3T + 5.71e5T^{2} \)
89 \( 1 + 86.8T + 7.04e5T^{2} \)
97 \( 1 - 758.T + 9.12e5T^{2} \)
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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

−11.83994496938007243824809703777, −10.59727566259860430950339573299, −9.255880147529164461980953078354, −8.289412217798938712708510784607, −7.55218325386562889401970212933, −6.41927456573235425930732721327, −4.61271525754772652377437242932, −4.12963763953552332174551385052, −3.04751035363347074690164446782, −2.02716598979314183859881085614, 2.02716598979314183859881085614, 3.04751035363347074690164446782, 4.12963763953552332174551385052, 4.61271525754772652377437242932, 6.41927456573235425930732721327, 7.55218325386562889401970212933, 8.289412217798938712708510784607, 9.255880147529164461980953078354, 10.59727566259860430950339573299, 11.83994496938007243824809703777

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