Properties

Label 2-17e2-1.1-c9-0-137
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  − 9.57·2-s − 3.76·3-s − 420.·4-s + 1.87e3·5-s + 36.0·6-s − 1.55e3·7-s + 8.92e3·8-s − 1.96e4·9-s − 1.79e4·10-s + 6.88e4·11-s + 1.58e3·12-s − 5.31e4·13-s + 1.48e4·14-s − 7.04e3·15-s + 1.29e5·16-s + 1.88e5·18-s + 6.43e4·19-s − 7.87e5·20-s + 5.84e3·21-s − 6.59e5·22-s − 2.42e6·23-s − 3.35e4·24-s + 1.55e6·25-s + 5.08e5·26-s + 1.48e5·27-s + 6.52e5·28-s + 4.67e6·29-s + ⋯
L(s)  = 1  − 0.423·2-s − 0.0268·3-s − 0.820·4-s + 1.34·5-s + 0.0113·6-s − 0.244·7-s + 0.770·8-s − 0.999·9-s − 0.567·10-s + 1.41·11-s + 0.0220·12-s − 0.515·13-s + 0.103·14-s − 0.0359·15-s + 0.494·16-s + 0.422·18-s + 0.113·19-s − 1.10·20-s + 0.00655·21-s − 0.599·22-s − 1.81·23-s − 0.0206·24-s + 0.796·25-s + 0.218·26-s + 0.0536·27-s + 0.200·28-s + 1.22·29-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: Trivial
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 \( 1 + 9.57T + 512T^{2} \)
3 \( 1 + 3.76T + 1.96e4T^{2} \)
5 \( 1 - 1.87e3T + 1.95e6T^{2} \)
7 \( 1 + 1.55e3T + 4.03e7T^{2} \)
11 \( 1 - 6.88e4T + 2.35e9T^{2} \)
13 \( 1 + 5.31e4T + 1.06e10T^{2} \)
19 \( 1 - 6.43e4T + 3.22e11T^{2} \)
23 \( 1 + 2.42e6T + 1.80e12T^{2} \)
29 \( 1 - 4.67e6T + 1.45e13T^{2} \)
31 \( 1 - 4.15e6T + 2.64e13T^{2} \)
37 \( 1 + 9.76e6T + 1.29e14T^{2} \)
41 \( 1 + 3.09e7T + 3.27e14T^{2} \)
43 \( 1 - 2.81e7T + 5.02e14T^{2} \)
47 \( 1 + 1.87e7T + 1.11e15T^{2} \)
53 \( 1 - 4.47e6T + 3.29e15T^{2} \)
59 \( 1 - 1.26e8T + 8.66e15T^{2} \)
61 \( 1 - 3.41e7T + 1.16e16T^{2} \)
67 \( 1 - 1.37e8T + 2.72e16T^{2} \)
71 \( 1 + 4.06e7T + 4.58e16T^{2} \)
73 \( 1 + 2.45e8T + 5.88e16T^{2} \)
79 \( 1 - 3.94e8T + 1.19e17T^{2} \)
83 \( 1 - 3.15e8T + 1.86e17T^{2} \)
89 \( 1 + 4.99e8T + 3.50e17T^{2} \)
97 \( 1 + 7.84e8T + 7.60e17T^{2} \)
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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

−9.796888315659731718037771764050, −8.917437455102680232085859107100, −8.208207591679858465611125509544, −6.65482816480763596597167465157, −5.87928793203451150395495970071, −4.87522103093538852341419163929, −3.63712408280707202669866851682, −2.24428703805061349956094750496, −1.19258587369311411653045247679, 0, 1.19258587369311411653045247679, 2.24428703805061349956094750496, 3.63712408280707202669866851682, 4.87522103093538852341419163929, 5.87928793203451150395495970071, 6.65482816480763596597167465157, 8.208207591679858465611125509544, 8.917437455102680232085859107100, 9.796888315659731718037771764050

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