Properties

 Label 2-17e2-1.1-c3-0-46 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 $1$

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Dirichlet series

 L(s)  = 1 + 2.68·2-s − 4.33·3-s − 0.785·4-s + 2.08·5-s − 11.6·6-s + 24.9·7-s − 23.5·8-s − 8.23·9-s + 5.60·10-s − 3.82·11-s + 3.40·12-s + 17.6·13-s + 67.1·14-s − 9.03·15-s − 57.1·16-s − 22.1·18-s − 160.·19-s − 1.63·20-s − 108.·21-s − 10.2·22-s − 99.9·23-s + 102.·24-s − 120.·25-s + 47.4·26-s + 152.·27-s − 19.6·28-s − 200.·29-s + ⋯
 L(s)  = 1 + 0.949·2-s − 0.833·3-s − 0.0981·4-s + 0.186·5-s − 0.791·6-s + 1.34·7-s − 1.04·8-s − 0.305·9-s + 0.177·10-s − 0.104·11-s + 0.0818·12-s + 0.377·13-s + 1.28·14-s − 0.155·15-s − 0.892·16-s − 0.289·18-s − 1.94·19-s − 0.0183·20-s − 1.12·21-s − 0.0994·22-s − 0.906·23-s + 0.869·24-s − 0.965·25-s + 0.358·26-s + 1.08·27-s − 0.132·28-s − 1.28·29-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: $$1$$ Selberg data: $$(2,\ 289,\ (\ :3/2),\ -1)$$

Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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 - 2.68T + 8T^{2}$$
3 $$1 + 4.33T + 27T^{2}$$
5 $$1 - 2.08T + 125T^{2}$$
7 $$1 - 24.9T + 343T^{2}$$
11 $$1 + 3.82T + 1.33e3T^{2}$$
13 $$1 - 17.6T + 2.19e3T^{2}$$
19 $$1 + 160.T + 6.85e3T^{2}$$
23 $$1 + 99.9T + 1.21e4T^{2}$$
29 $$1 + 200.T + 2.43e4T^{2}$$
31 $$1 - 76.5T + 2.97e4T^{2}$$
37 $$1 + 244.T + 5.06e4T^{2}$$
41 $$1 + 54.1T + 6.89e4T^{2}$$
43 $$1 - 142.T + 7.95e4T^{2}$$
47 $$1 + 468.T + 1.03e5T^{2}$$
53 $$1 + 96.5T + 1.48e5T^{2}$$
59 $$1 - 364.T + 2.05e5T^{2}$$
61 $$1 - 707.T + 2.26e5T^{2}$$
67 $$1 + 304.T + 3.00e5T^{2}$$
71 $$1 + 470.T + 3.57e5T^{2}$$
73 $$1 + 142.T + 3.89e5T^{2}$$
79 $$1 - 717.T + 4.93e5T^{2}$$
83 $$1 + 367.T + 5.71e5T^{2}$$
89 $$1 - 1.04e3T + 7.04e5T^{2}$$
97 $$1 - 903.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

−11.25663815305059234261978174745, −10.27913562657937888979597741929, −8.826694704173227030342047547956, −8.074251617472804068893413282024, −6.42928257209339339106637797026, −5.65572591429008564824069379719, −4.83204869234405139213720476123, −3.87838291463079950847578081580, −2.04184674760399630399440607845, 0, 2.04184674760399630399440607845, 3.87838291463079950847578081580, 4.83204869234405139213720476123, 5.65572591429008564824069379719, 6.42928257209339339106637797026, 8.074251617472804068893413282024, 8.826694704173227030342047547956, 10.27913562657937888979597741929, 11.25663815305059234261978174745