Properties

Label 2-17e2-1.1-c9-0-100
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 32.5·2-s − 213.·3-s + 547.·4-s + 2.40e3·5-s + 6.94e3·6-s + 8.34e3·7-s − 1.14e3·8-s + 2.58e4·9-s − 7.83e4·10-s − 2.94e4·11-s − 1.16e5·12-s + 1.14e5·13-s − 2.71e5·14-s − 5.13e5·15-s − 2.42e5·16-s − 8.40e5·18-s + 7.91e4·19-s + 1.31e6·20-s − 1.78e6·21-s + 9.57e5·22-s + 2.58e6·23-s + 2.43e5·24-s + 3.83e6·25-s − 3.73e6·26-s − 1.31e6·27-s + 4.56e6·28-s + 5.69e5·29-s + ⋯
L(s)  = 1  − 1.43·2-s − 1.52·3-s + 1.06·4-s + 1.72·5-s + 2.18·6-s + 1.31·7-s − 0.0986·8-s + 1.31·9-s − 2.47·10-s − 0.605·11-s − 1.62·12-s + 1.11·13-s − 1.89·14-s − 2.61·15-s − 0.926·16-s − 1.88·18-s + 0.139·19-s + 1.84·20-s − 1.99·21-s + 0.870·22-s + 1.92·23-s + 0.150·24-s + 1.96·25-s − 1.60·26-s − 0.475·27-s + 1.40·28-s + 0.149·29-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

\(L(5)\) \(\approx\) \(1.326583688\)
\(L(\frac12)\) \(\approx\) \(1.326583688\)
\(L(\frac{11}{2})\) 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 \( 1 + 32.5T + 512T^{2} \)
3 \( 1 + 213.T + 1.96e4T^{2} \)
5 \( 1 - 2.40e3T + 1.95e6T^{2} \)
7 \( 1 - 8.34e3T + 4.03e7T^{2} \)
11 \( 1 + 2.94e4T + 2.35e9T^{2} \)
13 \( 1 - 1.14e5T + 1.06e10T^{2} \)
19 \( 1 - 7.91e4T + 3.22e11T^{2} \)
23 \( 1 - 2.58e6T + 1.80e12T^{2} \)
29 \( 1 - 5.69e5T + 1.45e13T^{2} \)
31 \( 1 + 3.28e6T + 2.64e13T^{2} \)
37 \( 1 - 1.35e7T + 1.29e14T^{2} \)
41 \( 1 + 5.82e6T + 3.27e14T^{2} \)
43 \( 1 - 4.20e7T + 5.02e14T^{2} \)
47 \( 1 + 2.37e7T + 1.11e15T^{2} \)
53 \( 1 - 4.78e7T + 3.29e15T^{2} \)
59 \( 1 - 1.36e8T + 8.66e15T^{2} \)
61 \( 1 + 8.58e7T + 1.16e16T^{2} \)
67 \( 1 - 1.38e8T + 2.72e16T^{2} \)
71 \( 1 - 1.34e7T + 4.58e16T^{2} \)
73 \( 1 + 1.85e8T + 5.88e16T^{2} \)
79 \( 1 - 9.57e7T + 1.19e17T^{2} \)
83 \( 1 - 6.11e8T + 1.86e17T^{2} \)
89 \( 1 - 3.05e8T + 3.50e17T^{2} \)
97 \( 1 - 4.08e8T + 7.60e17T^{2} \)
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

−10.39400001368284580554172129162, −9.373521055293012143499686510188, −8.565287117377419016712680270042, −7.35366911619087766487120744843, −6.34311878086639010797666401672, −5.47144481751770327743851976741, −4.78279759438921814192076978784, −2.30484136642505250364266150015, −1.26165389292012219348114093638, −0.851096519639918461437654493234, 0.851096519639918461437654493234, 1.26165389292012219348114093638, 2.30484136642505250364266150015, 4.78279759438921814192076978784, 5.47144481751770327743851976741, 6.34311878086639010797666401672, 7.35366911619087766487120744843, 8.565287117377419016712680270042, 9.373521055293012143499686510188, 10.39400001368284580554172129162

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