Properties

Label 2-17e2-1.1-c9-0-127
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.9·2-s − 87.8·3-s − 369.·4-s − 13.1·5-s − 1.04e3·6-s + 9.37e3·7-s − 1.05e4·8-s − 1.19e4·9-s − 157.·10-s + 2.32e4·11-s + 3.24e4·12-s − 1.90e4·13-s + 1.12e5·14-s + 1.15e3·15-s + 6.32e4·16-s − 1.42e5·18-s − 4.94e5·19-s + 4.87e3·20-s − 8.23e5·21-s + 2.77e5·22-s − 1.11e6·23-s + 9.25e5·24-s − 1.95e6·25-s − 2.27e5·26-s + 2.78e6·27-s − 3.46e6·28-s + 5.04e6·29-s + ⋯
L(s)  = 1  + 0.528·2-s − 0.626·3-s − 0.721·4-s − 0.00944·5-s − 0.330·6-s + 1.47·7-s − 0.908·8-s − 0.607·9-s − 0.00498·10-s + 0.478·11-s + 0.451·12-s − 0.184·13-s + 0.779·14-s + 0.00591·15-s + 0.241·16-s − 0.321·18-s − 0.870·19-s + 0.00680·20-s − 0.924·21-s + 0.252·22-s − 0.830·23-s + 0.569·24-s − 0.999·25-s − 0.0974·26-s + 1.00·27-s − 1.06·28-s + 1.32·29-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: \(1\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 11.9T + 512T^{2} \)
3 \( 1 + 87.8T + 1.96e4T^{2} \)
5 \( 1 + 13.1T + 1.95e6T^{2} \)
7 \( 1 - 9.37e3T + 4.03e7T^{2} \)
11 \( 1 - 2.32e4T + 2.35e9T^{2} \)
13 \( 1 + 1.90e4T + 1.06e10T^{2} \)
19 \( 1 + 4.94e5T + 3.22e11T^{2} \)
23 \( 1 + 1.11e6T + 1.80e12T^{2} \)
29 \( 1 - 5.04e6T + 1.45e13T^{2} \)
31 \( 1 - 9.14e6T + 2.64e13T^{2} \)
37 \( 1 + 7.97e6T + 1.29e14T^{2} \)
41 \( 1 - 1.11e7T + 3.27e14T^{2} \)
43 \( 1 + 4.06e6T + 5.02e14T^{2} \)
47 \( 1 - 1.71e6T + 1.11e15T^{2} \)
53 \( 1 - 4.70e6T + 3.29e15T^{2} \)
59 \( 1 - 1.62e8T + 8.66e15T^{2} \)
61 \( 1 + 8.09e7T + 1.16e16T^{2} \)
67 \( 1 + 1.08e8T + 2.72e16T^{2} \)
71 \( 1 - 2.89e8T + 4.58e16T^{2} \)
73 \( 1 - 8.83e7T + 5.88e16T^{2} \)
79 \( 1 + 5.77e8T + 1.19e17T^{2} \)
83 \( 1 - 4.14e8T + 1.86e17T^{2} \)
89 \( 1 + 7.23e8T + 3.50e17T^{2} \)
97 \( 1 + 3.43e8T + 7.60e17T^{2} \)
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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

−9.856159385450493598081593145074, −8.536860367616787607327716568886, −8.144582024627656215686804309897, −6.49743617127093589757792846185, −5.60573376272900334255332651485, −4.74957218624707420212534206412, −4.05635847690737305568589832191, −2.51902789910477223272114444392, −1.12712223772286017652675325117, 0, 1.12712223772286017652675325117, 2.51902789910477223272114444392, 4.05635847690737305568589832191, 4.74957218624707420212534206412, 5.60573376272900334255332651485, 6.49743617127093589757792846185, 8.144582024627656215686804309897, 8.536860367616787607327716568886, 9.856159385450493598081593145074

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