Properties

Label 2-17e2-1.1-c9-0-91
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  + 19.5·2-s − 220.·3-s − 128.·4-s + 130.·5-s − 4.31e3·6-s − 9.77e3·7-s − 1.25e4·8-s + 2.87e4·9-s + 2.54e3·10-s − 7.12e4·11-s + 2.82e4·12-s + 1.26e5·13-s − 1.91e5·14-s − 2.86e4·15-s − 1.79e5·16-s + 5.63e5·18-s + 6.04e5·19-s − 1.67e4·20-s + 2.15e6·21-s − 1.39e6·22-s + 1.07e6·23-s + 2.76e6·24-s − 1.93e6·25-s + 2.48e6·26-s − 1.99e6·27-s + 1.25e6·28-s + 1.91e6·29-s + ⋯
L(s)  = 1  + 0.865·2-s − 1.56·3-s − 0.250·4-s + 0.0930·5-s − 1.35·6-s − 1.53·7-s − 1.08·8-s + 1.46·9-s + 0.0805·10-s − 1.46·11-s + 0.393·12-s + 1.23·13-s − 1.33·14-s − 0.145·15-s − 0.686·16-s + 1.26·18-s + 1.06·19-s − 0.0233·20-s + 2.41·21-s − 1.27·22-s + 0.801·23-s + 1.69·24-s − 0.991·25-s + 1.06·26-s − 0.722·27-s + 0.385·28-s + 0.502·29-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 \( 1 - 19.5T + 512T^{2} \)
3 \( 1 + 220.T + 1.96e4T^{2} \)
5 \( 1 - 130.T + 1.95e6T^{2} \)
7 \( 1 + 9.77e3T + 4.03e7T^{2} \)
11 \( 1 + 7.12e4T + 2.35e9T^{2} \)
13 \( 1 - 1.26e5T + 1.06e10T^{2} \)
19 \( 1 - 6.04e5T + 3.22e11T^{2} \)
23 \( 1 - 1.07e6T + 1.80e12T^{2} \)
29 \( 1 - 1.91e6T + 1.45e13T^{2} \)
31 \( 1 + 7.71e6T + 2.64e13T^{2} \)
37 \( 1 - 1.53e7T + 1.29e14T^{2} \)
41 \( 1 - 1.94e7T + 3.27e14T^{2} \)
43 \( 1 + 1.95e7T + 5.02e14T^{2} \)
47 \( 1 - 4.08e7T + 1.11e15T^{2} \)
53 \( 1 + 4.89e5T + 3.29e15T^{2} \)
59 \( 1 - 1.93e7T + 8.66e15T^{2} \)
61 \( 1 + 9.43e7T + 1.16e16T^{2} \)
67 \( 1 + 1.30e8T + 2.72e16T^{2} \)
71 \( 1 - 1.90e8T + 4.58e16T^{2} \)
73 \( 1 + 8.01e7T + 5.88e16T^{2} \)
79 \( 1 + 4.53e8T + 1.19e17T^{2} \)
83 \( 1 - 2.97e8T + 1.86e17T^{2} \)
89 \( 1 - 7.19e8T + 3.50e17T^{2} \)
97 \( 1 + 2.68e8T + 7.60e17T^{2} \)
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

−9.976363825621525583417791297671, −9.069822396590216040822903048184, −7.45503889596285939865506933416, −6.21837960910209954297996656291, −5.80876922286489458549893422266, −5.02713900771907176948200504619, −3.81621122897883128827506776240, −2.86771731264720734824303349982, −0.804195963514882035689539386181, 0, 0.804195963514882035689539386181, 2.86771731264720734824303349982, 3.81621122897883128827506776240, 5.02713900771907176948200504619, 5.80876922286489458549893422266, 6.21837960910209954297996656291, 7.45503889596285939865506933416, 9.069822396590216040822903048184, 9.976363825621525583417791297671

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