Properties

Label 2-17e2-1.1-c9-0-140
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  + 19.0·2-s − 138.·3-s − 148.·4-s + 952.·5-s − 2.63e3·6-s + 6.30e3·7-s − 1.25e4·8-s − 541.·9-s + 1.81e4·10-s + 1.81e4·11-s + 2.06e4·12-s − 2.24e4·13-s + 1.20e5·14-s − 1.31e5·15-s − 1.63e5·16-s − 1.03e4·18-s + 1.67e5·19-s − 1.41e5·20-s − 8.72e5·21-s + 3.45e5·22-s − 2.37e5·23-s + 1.74e6·24-s − 1.04e6·25-s − 4.28e5·26-s + 2.79e6·27-s − 9.39e5·28-s − 7.21e6·29-s + ⋯
L(s)  = 1  + 0.842·2-s − 0.986·3-s − 0.290·4-s + 0.681·5-s − 0.830·6-s + 0.992·7-s − 1.08·8-s − 0.0275·9-s + 0.574·10-s + 0.373·11-s + 0.286·12-s − 0.218·13-s + 0.835·14-s − 0.672·15-s − 0.624·16-s − 0.0231·18-s + 0.295·19-s − 0.198·20-s − 0.978·21-s + 0.314·22-s − 0.176·23-s + 1.07·24-s − 0.535·25-s − 0.183·26-s + 1.01·27-s − 0.288·28-s − 1.89·29-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 \( 1 - 19.0T + 512T^{2} \)
3 \( 1 + 138.T + 1.96e4T^{2} \)
5 \( 1 - 952.T + 1.95e6T^{2} \)
7 \( 1 - 6.30e3T + 4.03e7T^{2} \)
11 \( 1 - 1.81e4T + 2.35e9T^{2} \)
13 \( 1 + 2.24e4T + 1.06e10T^{2} \)
19 \( 1 - 1.67e5T + 3.22e11T^{2} \)
23 \( 1 + 2.37e5T + 1.80e12T^{2} \)
29 \( 1 + 7.21e6T + 1.45e13T^{2} \)
31 \( 1 - 7.65e6T + 2.64e13T^{2} \)
37 \( 1 - 5.52e6T + 1.29e14T^{2} \)
41 \( 1 - 5.44e6T + 3.27e14T^{2} \)
43 \( 1 - 2.86e7T + 5.02e14T^{2} \)
47 \( 1 - 1.86e7T + 1.11e15T^{2} \)
53 \( 1 + 3.78e6T + 3.29e15T^{2} \)
59 \( 1 + 9.03e7T + 8.66e15T^{2} \)
61 \( 1 + 6.48e7T + 1.16e16T^{2} \)
67 \( 1 - 3.00e8T + 2.72e16T^{2} \)
71 \( 1 + 9.27e7T + 4.58e16T^{2} \)
73 \( 1 + 3.79e8T + 5.88e16T^{2} \)
79 \( 1 - 2.94e8T + 1.19e17T^{2} \)
83 \( 1 + 5.73e8T + 1.86e17T^{2} \)
89 \( 1 - 3.06e8T + 3.50e17T^{2} \)
97 \( 1 + 2.52e8T + 7.60e17T^{2} \)
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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

−9.816211276773553350068082450219, −8.931447507236990877721254296043, −7.73028342789724553714780905794, −6.28136734630386133391581053062, −5.66325987879637796199541079756, −4.92867594432478919504237675197, −4.01862182693380640452352881223, −2.54466259570274231448980499740, −1.21706702242832168016490759924, 0, 1.21706702242832168016490759924, 2.54466259570274231448980499740, 4.01862182693380640452352881223, 4.92867594432478919504237675197, 5.66325987879637796199541079756, 6.28136734630386133391581053062, 7.73028342789724553714780905794, 8.931447507236990877721254296043, 9.816211276773553350068082450219

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