Properties

Label 2-17e2-1.1-c9-0-102
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  − 22.6·2-s − 265.·3-s + 1.89·4-s − 368.·5-s + 6.02e3·6-s + 2.87e3·7-s + 1.15e4·8-s + 5.08e4·9-s + 8.34e3·10-s + 6.38e4·11-s − 503.·12-s − 1.49e5·13-s − 6.52e4·14-s + 9.77e4·15-s − 2.63e5·16-s − 1.15e6·18-s + 4.44e5·19-s − 698.·20-s − 7.64e5·21-s − 1.44e6·22-s + 1.56e6·23-s − 3.07e6·24-s − 1.81e6·25-s + 3.38e6·26-s − 8.28e6·27-s + 5.45e3·28-s − 5.40e6·29-s + ⋯
L(s)  = 1  − 1.00·2-s − 1.89·3-s + 0.00370·4-s − 0.263·5-s + 1.89·6-s + 0.452·7-s + 0.998·8-s + 2.58·9-s + 0.263·10-s + 1.31·11-s − 0.00701·12-s − 1.44·13-s − 0.453·14-s + 0.498·15-s − 1.00·16-s − 2.58·18-s + 0.782·19-s − 0.000975·20-s − 0.857·21-s − 1.31·22-s + 1.16·23-s − 1.88·24-s − 0.930·25-s + 1.45·26-s − 2.99·27-s + 0.00167·28-s − 1.41·29-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 \( 1 + 22.6T + 512T^{2} \)
3 \( 1 + 265.T + 1.96e4T^{2} \)
5 \( 1 + 368.T + 1.95e6T^{2} \)
7 \( 1 - 2.87e3T + 4.03e7T^{2} \)
11 \( 1 - 6.38e4T + 2.35e9T^{2} \)
13 \( 1 + 1.49e5T + 1.06e10T^{2} \)
19 \( 1 - 4.44e5T + 3.22e11T^{2} \)
23 \( 1 - 1.56e6T + 1.80e12T^{2} \)
29 \( 1 + 5.40e6T + 1.45e13T^{2} \)
31 \( 1 + 7.59e5T + 2.64e13T^{2} \)
37 \( 1 + 3.93e6T + 1.29e14T^{2} \)
41 \( 1 - 9.97e6T + 3.27e14T^{2} \)
43 \( 1 - 2.04e7T + 5.02e14T^{2} \)
47 \( 1 - 4.76e7T + 1.11e15T^{2} \)
53 \( 1 + 5.80e6T + 3.29e15T^{2} \)
59 \( 1 - 9.50e7T + 8.66e15T^{2} \)
61 \( 1 + 1.59e8T + 1.16e16T^{2} \)
67 \( 1 - 8.38e7T + 2.72e16T^{2} \)
71 \( 1 + 3.10e8T + 4.58e16T^{2} \)
73 \( 1 + 3.37e8T + 5.88e16T^{2} \)
79 \( 1 - 5.51e7T + 1.19e17T^{2} \)
83 \( 1 - 3.47e8T + 1.86e17T^{2} \)
89 \( 1 + 3.74e8T + 3.50e17T^{2} \)
97 \( 1 + 1.18e9T + 7.60e17T^{2} \)
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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

−9.758318044008316106930372845175, −9.161616630974150374543416511220, −7.47489768875469794434537530353, −7.16451383958968313517320182242, −5.79606862568682495090640838916, −4.87364564852129388382764509734, −4.08865398190369719817645107841, −1.70020331940169049542753719642, −0.860076666186462442866622812350, 0, 0.860076666186462442866622812350, 1.70020331940169049542753719642, 4.08865398190369719817645107841, 4.87364564852129388382764509734, 5.79606862568682495090640838916, 7.16451383958968313517320182242, 7.47489768875469794434537530353, 9.161616630974150374543416511220, 9.758318044008316106930372845175

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