Properties

Label 2-17e2-1.1-c9-0-123
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 37.7·2-s − 226.·3-s + 910.·4-s − 2.06e3·5-s − 8.53e3·6-s − 6.87e3·7-s + 1.50e4·8-s + 3.15e4·9-s − 7.77e4·10-s + 6.14e4·11-s − 2.06e5·12-s + 1.45e5·13-s − 2.59e5·14-s + 4.66e5·15-s + 1.00e5·16-s + 1.18e6·18-s − 6.68e5·19-s − 1.87e6·20-s + 1.55e6·21-s + 2.31e6·22-s − 4.70e5·23-s − 3.40e6·24-s + 2.29e6·25-s + 5.47e6·26-s − 2.67e6·27-s − 6.25e6·28-s + 5.12e6·29-s + ⋯
L(s)  = 1  + 1.66·2-s − 1.61·3-s + 1.77·4-s − 1.47·5-s − 2.68·6-s − 1.08·7-s + 1.29·8-s + 1.60·9-s − 2.45·10-s + 1.26·11-s − 2.86·12-s + 1.41·13-s − 1.80·14-s + 2.37·15-s + 0.385·16-s + 2.66·18-s − 1.17·19-s − 2.62·20-s + 1.74·21-s + 2.10·22-s − 0.350·23-s − 2.09·24-s + 1.17·25-s + 2.35·26-s − 0.968·27-s − 1.92·28-s + 1.34·29-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 \( 1 - 37.7T + 512T^{2} \)
3 \( 1 + 226.T + 1.96e4T^{2} \)
5 \( 1 + 2.06e3T + 1.95e6T^{2} \)
7 \( 1 + 6.87e3T + 4.03e7T^{2} \)
11 \( 1 - 6.14e4T + 2.35e9T^{2} \)
13 \( 1 - 1.45e5T + 1.06e10T^{2} \)
19 \( 1 + 6.68e5T + 3.22e11T^{2} \)
23 \( 1 + 4.70e5T + 1.80e12T^{2} \)
29 \( 1 - 5.12e6T + 1.45e13T^{2} \)
31 \( 1 - 4.67e6T + 2.64e13T^{2} \)
37 \( 1 - 6.72e6T + 1.29e14T^{2} \)
41 \( 1 - 1.09e7T + 3.27e14T^{2} \)
43 \( 1 + 7.03e6T + 5.02e14T^{2} \)
47 \( 1 - 4.68e7T + 1.11e15T^{2} \)
53 \( 1 - 2.76e7T + 3.29e15T^{2} \)
59 \( 1 + 7.22e7T + 8.66e15T^{2} \)
61 \( 1 + 8.21e7T + 1.16e16T^{2} \)
67 \( 1 + 1.82e8T + 2.72e16T^{2} \)
71 \( 1 - 2.28e7T + 4.58e16T^{2} \)
73 \( 1 + 3.41e8T + 5.88e16T^{2} \)
79 \( 1 + 1.65e8T + 1.19e17T^{2} \)
83 \( 1 + 8.05e8T + 1.86e17T^{2} \)
89 \( 1 + 5.38e8T + 3.50e17T^{2} \)
97 \( 1 + 3.81e8T + 7.60e17T^{2} \)
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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

−10.40441299073455670964540181985, −8.697246675152642786504851415646, −7.10990631233084100538946051610, −6.27414373898285200571766864941, −6.04031229898044937060795900961, −4.37952879972107388834441917181, −4.17210356925158909447156533384, −3.11961652784173738921832112071, −1.08669818420129574852231086414, 0, 1.08669818420129574852231086414, 3.11961652784173738921832112071, 4.17210356925158909447156533384, 4.37952879972107388834441917181, 6.04031229898044937060795900961, 6.27414373898285200571766864941, 7.10990631233084100538946051610, 8.697246675152642786504851415646, 10.40441299073455670964540181985

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