Properties

Label 2-17e2-1.1-c9-0-48
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.7·2-s + 53.3·3-s − 396.·4-s − 85.7·5-s + 572.·6-s + 4.51e3·7-s − 9.74e3·8-s − 1.68e4·9-s − 919.·10-s − 3.15e4·11-s − 2.11e4·12-s − 6.48e3·13-s + 4.83e4·14-s − 4.57e3·15-s + 9.86e4·16-s − 1.80e5·18-s + 5.08e5·19-s + 3.40e4·20-s + 2.40e5·21-s − 3.38e5·22-s − 1.78e6·23-s − 5.20e5·24-s − 1.94e6·25-s − 6.95e4·26-s − 1.94e6·27-s − 1.79e6·28-s − 1.55e6·29-s + ⋯
L(s)  = 1  + 0.473·2-s + 0.380·3-s − 0.775·4-s − 0.0613·5-s + 0.180·6-s + 0.710·7-s − 0.841·8-s − 0.855·9-s − 0.0290·10-s − 0.649·11-s − 0.294·12-s − 0.0629·13-s + 0.336·14-s − 0.0233·15-s + 0.376·16-s − 0.405·18-s + 0.894·19-s + 0.0475·20-s + 0.270·21-s − 0.307·22-s − 1.33·23-s − 0.320·24-s − 0.996·25-s − 0.0298·26-s − 0.705·27-s − 0.550·28-s − 0.408·29-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: \(0\)
Selberg data: \((2,\ 289,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.696578533\)
\(L(\frac12)\) \(\approx\) \(1.696578533\)
\(L(\frac{11}{2})\) 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 \( 1 - 10.7T + 512T^{2} \)
3 \( 1 - 53.3T + 1.96e4T^{2} \)
5 \( 1 + 85.7T + 1.95e6T^{2} \)
7 \( 1 - 4.51e3T + 4.03e7T^{2} \)
11 \( 1 + 3.15e4T + 2.35e9T^{2} \)
13 \( 1 + 6.48e3T + 1.06e10T^{2} \)
19 \( 1 - 5.08e5T + 3.22e11T^{2} \)
23 \( 1 + 1.78e6T + 1.80e12T^{2} \)
29 \( 1 + 1.55e6T + 1.45e13T^{2} \)
31 \( 1 - 2.05e6T + 2.64e13T^{2} \)
37 \( 1 + 7.61e6T + 1.29e14T^{2} \)
41 \( 1 - 5.31e6T + 3.27e14T^{2} \)
43 \( 1 - 2.91e7T + 5.02e14T^{2} \)
47 \( 1 - 1.71e7T + 1.11e15T^{2} \)
53 \( 1 + 4.87e7T + 3.29e15T^{2} \)
59 \( 1 + 1.48e7T + 8.66e15T^{2} \)
61 \( 1 - 1.92e8T + 1.16e16T^{2} \)
67 \( 1 - 3.41e7T + 2.72e16T^{2} \)
71 \( 1 + 1.63e8T + 4.58e16T^{2} \)
73 \( 1 - 8.46e7T + 5.88e16T^{2} \)
79 \( 1 + 3.66e8T + 1.19e17T^{2} \)
83 \( 1 - 5.02e8T + 1.86e17T^{2} \)
89 \( 1 - 8.00e7T + 3.50e17T^{2} \)
97 \( 1 - 1.90e8T + 7.60e17T^{2} \)
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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

−10.10422171016759049733235666435, −9.211780015141880621945917794568, −8.254662334357917775680762838547, −7.67685163846715650054096827750, −5.93519257252028551952376208352, −5.27483282411022668200173498310, −4.19663626893381301018873831155, −3.21134561607800624211111925823, −2.07508701201550633680250781956, −0.51365744167432124586909841718, 0.51365744167432124586909841718, 2.07508701201550633680250781956, 3.21134561607800624211111925823, 4.19663626893381301018873831155, 5.27483282411022668200173498310, 5.93519257252028551952376208352, 7.67685163846715650054096827750, 8.254662334357917775680762838547, 9.211780015141880621945917794568, 10.10422171016759049733235666435

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