Properties

Label 2-17e2-1.1-c9-0-61
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  − 3.07·2-s + 18.6·3-s − 502.·4-s − 2.54e3·5-s − 57.2·6-s − 1.09e4·7-s + 3.11e3·8-s − 1.93e4·9-s + 7.80e3·10-s + 5.50e4·11-s − 9.35e3·12-s − 1.01e5·13-s + 3.36e4·14-s − 4.72e4·15-s + 2.47e5·16-s + 5.94e4·18-s − 2.43e5·19-s + 1.27e6·20-s − 2.03e5·21-s − 1.69e5·22-s − 7.99e5·23-s + 5.80e4·24-s + 4.49e6·25-s + 3.13e5·26-s − 7.26e5·27-s + 5.50e6·28-s + 1.08e6·29-s + ⋯
L(s)  = 1  − 0.135·2-s + 0.132·3-s − 0.981·4-s − 1.81·5-s − 0.0180·6-s − 1.72·7-s + 0.269·8-s − 0.982·9-s + 0.246·10-s + 1.13·11-s − 0.130·12-s − 0.989·13-s + 0.234·14-s − 0.241·15-s + 0.944·16-s + 0.133·18-s − 0.428·19-s + 1.78·20-s − 0.228·21-s − 0.153·22-s − 0.595·23-s + 0.0357·24-s + 2.30·25-s + 0.134·26-s − 0.262·27-s + 1.69·28-s + 0.285·29-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 \( 1 + 3.07T + 512T^{2} \)
3 \( 1 - 18.6T + 1.96e4T^{2} \)
5 \( 1 + 2.54e3T + 1.95e6T^{2} \)
7 \( 1 + 1.09e4T + 4.03e7T^{2} \)
11 \( 1 - 5.50e4T + 2.35e9T^{2} \)
13 \( 1 + 1.01e5T + 1.06e10T^{2} \)
19 \( 1 + 2.43e5T + 3.22e11T^{2} \)
23 \( 1 + 7.99e5T + 1.80e12T^{2} \)
29 \( 1 - 1.08e6T + 1.45e13T^{2} \)
31 \( 1 + 6.11e6T + 2.64e13T^{2} \)
37 \( 1 - 1.09e7T + 1.29e14T^{2} \)
41 \( 1 - 2.74e6T + 3.27e14T^{2} \)
43 \( 1 - 3.55e7T + 5.02e14T^{2} \)
47 \( 1 + 1.19e7T + 1.11e15T^{2} \)
53 \( 1 - 6.12e7T + 3.29e15T^{2} \)
59 \( 1 + 7.89e7T + 8.66e15T^{2} \)
61 \( 1 - 5.32e7T + 1.16e16T^{2} \)
67 \( 1 + 1.84e8T + 2.72e16T^{2} \)
71 \( 1 + 1.25e8T + 4.58e16T^{2} \)
73 \( 1 - 9.89e6T + 5.88e16T^{2} \)
79 \( 1 + 3.48e8T + 1.19e17T^{2} \)
83 \( 1 + 1.63e8T + 1.86e17T^{2} \)
89 \( 1 - 1.08e9T + 3.50e17T^{2} \)
97 \( 1 + 1.20e8T + 7.60e17T^{2} \)
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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

−9.437074787992650310723401064980, −8.939914670510647004651803926072, −7.926809090072289849998915696108, −7.02581745246938906133509845072, −5.85740740253883531413032834706, −4.35044356377462712821905127414, −3.74073055574973362091398739532, −2.88570370130235083720936653788, −0.61189935441812234227697039555, 0, 0.61189935441812234227697039555, 2.88570370130235083720936653788, 3.74073055574973362091398739532, 4.35044356377462712821905127414, 5.85740740253883531413032834706, 7.02581745246938906133509845072, 7.926809090072289849998915696108, 8.939914670510647004651803926072, 9.437074787992650310723401064980

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