Properties

Label 2-17e2-1.1-c9-0-120
Degree $2$
Conductor $289$
Sign $1$
Analytic cond. $148.845$
Root an. cond. $12.2002$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 40.9·2-s − 105.·3-s + 1.16e3·4-s + 2.07e3·5-s − 4.30e3·6-s + 5.18e3·7-s + 2.67e4·8-s − 8.65e3·9-s + 8.48e4·10-s − 8.39e4·11-s − 1.22e5·12-s − 3.84e4·13-s + 2.12e5·14-s − 2.17e5·15-s + 4.99e5·16-s − 3.54e5·18-s + 7.27e5·19-s + 2.41e6·20-s − 5.44e5·21-s − 3.43e6·22-s + 2.28e6·23-s − 2.81e6·24-s + 2.34e6·25-s − 1.57e6·26-s + 2.97e6·27-s + 6.03e6·28-s + 2.05e6·29-s + ⋯
L(s)  = 1  + 1.81·2-s − 0.748·3-s + 2.27·4-s + 1.48·5-s − 1.35·6-s + 0.815·7-s + 2.31·8-s − 0.439·9-s + 2.68·10-s − 1.72·11-s − 1.70·12-s − 0.373·13-s + 1.47·14-s − 1.11·15-s + 1.90·16-s − 0.795·18-s + 1.28·19-s + 3.37·20-s − 0.610·21-s − 3.12·22-s + 1.70·23-s − 1.72·24-s + 1.19·25-s − 0.675·26-s + 1.07·27-s + 1.85·28-s + 0.540·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $1$
Analytic conductor: \(148.845\)
Root analytic conductor: \(12.2002\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(8.282504765\)
\(L(\frac12)\) \(\approx\) \(8.282504765\)
\(L(\frac{11}{2})\) 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 \( 1 - 40.9T + 512T^{2} \)
3 \( 1 + 105.T + 1.96e4T^{2} \)
5 \( 1 - 2.07e3T + 1.95e6T^{2} \)
7 \( 1 - 5.18e3T + 4.03e7T^{2} \)
11 \( 1 + 8.39e4T + 2.35e9T^{2} \)
13 \( 1 + 3.84e4T + 1.06e10T^{2} \)
19 \( 1 - 7.27e5T + 3.22e11T^{2} \)
23 \( 1 - 2.28e6T + 1.80e12T^{2} \)
29 \( 1 - 2.05e6T + 1.45e13T^{2} \)
31 \( 1 - 2.10e6T + 2.64e13T^{2} \)
37 \( 1 - 1.39e7T + 1.29e14T^{2} \)
41 \( 1 - 2.53e7T + 3.27e14T^{2} \)
43 \( 1 + 2.22e7T + 5.02e14T^{2} \)
47 \( 1 - 1.95e7T + 1.11e15T^{2} \)
53 \( 1 + 3.74e6T + 3.29e15T^{2} \)
59 \( 1 + 7.17e7T + 8.66e15T^{2} \)
61 \( 1 + 1.17e7T + 1.16e16T^{2} \)
67 \( 1 - 3.00e8T + 2.72e16T^{2} \)
71 \( 1 - 1.31e8T + 4.58e16T^{2} \)
73 \( 1 - 3.21e8T + 5.88e16T^{2} \)
79 \( 1 - 3.05e8T + 1.19e17T^{2} \)
83 \( 1 + 4.83e8T + 1.86e17T^{2} \)
89 \( 1 - 1.76e8T + 3.50e17T^{2} \)
97 \( 1 + 8.15e8T + 7.60e17T^{2} \)
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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

−10.79717731791838495551369101905, −9.634016947097066568336542316592, −7.947702625630327412128697876635, −6.78826382967578988996691920986, −5.78858867314542138823376894064, −5.19518099523936453397497388617, −4.86242929731295570155059654575, −2.91228539917387576748400479566, −2.40175230088651581260735481560, −1.04729490711173885427571254659, 1.04729490711173885427571254659, 2.40175230088651581260735481560, 2.91228539917387576748400479566, 4.86242929731295570155059654575, 5.19518099523936453397497388617, 5.78858867314542138823376894064, 6.78826382967578988996691920986, 7.947702625630327412128697876635, 9.634016947097066568336542316592, 10.79717731791838495551369101905

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