Properties

Label 2-17e2-1.1-c9-0-65
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  + 29.4·2-s − 172.·3-s + 352.·4-s + 865.·5-s − 5.06e3·6-s + 362.·7-s − 4.67e3·8-s + 9.98e3·9-s + 2.54e4·10-s − 2.58e3·11-s − 6.07e4·12-s + 1.51e5·13-s + 1.06e4·14-s − 1.49e5·15-s − 3.18e5·16-s + 2.93e5·18-s − 6.14e5·19-s + 3.05e5·20-s − 6.23e4·21-s − 7.60e4·22-s − 9.25e4·23-s + 8.05e5·24-s − 1.20e6·25-s + 4.46e6·26-s + 1.67e6·27-s + 1.27e5·28-s + 4.01e6·29-s + ⋯
L(s)  = 1  + 1.29·2-s − 1.22·3-s + 0.689·4-s + 0.619·5-s − 1.59·6-s + 0.0570·7-s − 0.403·8-s + 0.507·9-s + 0.805·10-s − 0.0532·11-s − 0.846·12-s + 1.47·13-s + 0.0741·14-s − 0.760·15-s − 1.21·16-s + 0.659·18-s − 1.08·19-s + 0.426·20-s − 0.0700·21-s − 0.0691·22-s − 0.0689·23-s + 0.495·24-s − 0.616·25-s + 1.91·26-s + 0.605·27-s + 0.0393·28-s + 1.05·29-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

\(L(5)\) \(\approx\) \(2.670686068\)
\(L(\frac12)\) \(\approx\) \(2.670686068\)
\(L(\frac{11}{2})\) 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 \( 1 - 29.4T + 512T^{2} \)
3 \( 1 + 172.T + 1.96e4T^{2} \)
5 \( 1 - 865.T + 1.95e6T^{2} \)
7 \( 1 - 362.T + 4.03e7T^{2} \)
11 \( 1 + 2.58e3T + 2.35e9T^{2} \)
13 \( 1 - 1.51e5T + 1.06e10T^{2} \)
19 \( 1 + 6.14e5T + 3.22e11T^{2} \)
23 \( 1 + 9.25e4T + 1.80e12T^{2} \)
29 \( 1 - 4.01e6T + 1.45e13T^{2} \)
31 \( 1 + 1.53e6T + 2.64e13T^{2} \)
37 \( 1 + 1.19e7T + 1.29e14T^{2} \)
41 \( 1 - 2.69e7T + 3.27e14T^{2} \)
43 \( 1 - 1.73e7T + 5.02e14T^{2} \)
47 \( 1 + 2.76e7T + 1.11e15T^{2} \)
53 \( 1 + 5.68e7T + 3.29e15T^{2} \)
59 \( 1 + 4.42e7T + 8.66e15T^{2} \)
61 \( 1 + 1.39e8T + 1.16e16T^{2} \)
67 \( 1 - 2.98e8T + 2.72e16T^{2} \)
71 \( 1 - 2.46e8T + 4.58e16T^{2} \)
73 \( 1 - 4.68e8T + 5.88e16T^{2} \)
79 \( 1 - 3.27e8T + 1.19e17T^{2} \)
83 \( 1 - 2.96e8T + 1.86e17T^{2} \)
89 \( 1 - 5.63e7T + 3.50e17T^{2} \)
97 \( 1 - 9.04e8T + 7.60e17T^{2} \)
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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

−10.76866437426702523614850274253, −9.426069570000244243733521169839, −8.290555307782785157306739124917, −6.47032674662220290676430789222, −6.20430161006970191792926547618, −5.32950591753333004291629489556, −4.46159654002851542879459708360, −3.39767400278521836815271611083, −2.00069803329177686252501348911, −0.62690739114490293528887677848, 0.62690739114490293528887677848, 2.00069803329177686252501348911, 3.39767400278521836815271611083, 4.46159654002851542879459708360, 5.32950591753333004291629489556, 6.20430161006970191792926547618, 6.47032674662220290676430789222, 8.290555307782785157306739124917, 9.426069570000244243733521169839, 10.76866437426702523614850274253

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