Properties

Label 2-17e2-1.1-c9-0-135
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 38.3·2-s + 247.·3-s + 958.·4-s − 875.·5-s − 9.47e3·6-s − 7.07e3·7-s − 1.71e4·8-s + 4.13e4·9-s + 3.35e4·10-s + 5.52e4·11-s + 2.36e5·12-s − 1.79e5·13-s + 2.71e5·14-s − 2.16e5·15-s + 1.65e5·16-s − 1.58e6·18-s − 5.65e4·19-s − 8.39e5·20-s − 1.74e6·21-s − 2.12e6·22-s + 2.36e6·23-s − 4.22e6·24-s − 1.18e6·25-s + 6.89e6·26-s + 5.35e6·27-s − 6.77e6·28-s − 1.38e6·29-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.76·3-s + 1.87·4-s − 0.626·5-s − 2.98·6-s − 1.11·7-s − 1.47·8-s + 2.10·9-s + 1.06·10-s + 1.13·11-s + 3.29·12-s − 1.74·13-s + 1.88·14-s − 1.10·15-s + 0.631·16-s − 3.55·18-s − 0.0994·19-s − 1.17·20-s − 1.96·21-s − 1.92·22-s + 1.76·23-s − 2.60·24-s − 0.607·25-s + 2.95·26-s + 1.93·27-s − 2.08·28-s − 0.364·29-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 \( 1 + 38.3T + 512T^{2} \)
3 \( 1 - 247.T + 1.96e4T^{2} \)
5 \( 1 + 875.T + 1.95e6T^{2} \)
7 \( 1 + 7.07e3T + 4.03e7T^{2} \)
11 \( 1 - 5.52e4T + 2.35e9T^{2} \)
13 \( 1 + 1.79e5T + 1.06e10T^{2} \)
19 \( 1 + 5.65e4T + 3.22e11T^{2} \)
23 \( 1 - 2.36e6T + 1.80e12T^{2} \)
29 \( 1 + 1.38e6T + 1.45e13T^{2} \)
31 \( 1 - 4.30e5T + 2.64e13T^{2} \)
37 \( 1 - 1.73e7T + 1.29e14T^{2} \)
41 \( 1 - 2.41e7T + 3.27e14T^{2} \)
43 \( 1 + 3.24e6T + 5.02e14T^{2} \)
47 \( 1 + 4.13e7T + 1.11e15T^{2} \)
53 \( 1 - 1.71e7T + 3.29e15T^{2} \)
59 \( 1 + 3.33e7T + 8.66e15T^{2} \)
61 \( 1 - 5.58e7T + 1.16e16T^{2} \)
67 \( 1 + 1.25e8T + 2.72e16T^{2} \)
71 \( 1 + 6.66e7T + 4.58e16T^{2} \)
73 \( 1 + 2.13e6T + 5.88e16T^{2} \)
79 \( 1 + 3.27e8T + 1.19e17T^{2} \)
83 \( 1 - 2.63e8T + 1.86e17T^{2} \)
89 \( 1 - 1.86e8T + 3.50e17T^{2} \)
97 \( 1 + 9.45e7T + 7.60e17T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.420962284204280275349508088292, −9.078594691266352076694775912577, −7.975021332717585274590252059235, −7.34119892775865699120568776405, −6.66222514181203900744464399512, −4.28231319644359085878324862753, −3.10665117090072926172767232497, −2.40497831593821909154744088300, −1.18556978115202356238490335698, 0, 1.18556978115202356238490335698, 2.40497831593821909154744088300, 3.10665117090072926172767232497, 4.28231319644359085878324862753, 6.66222514181203900744464399512, 7.34119892775865699120568776405, 7.975021332717585274590252059235, 9.078594691266352076694775912577, 9.420962284204280275349508088292

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