Properties

Label 2-17e2-1.1-c9-0-125
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  − 10.3·2-s + 121.·3-s − 404.·4-s − 1.45e3·5-s − 1.26e3·6-s + 5.78e3·7-s + 9.51e3·8-s − 4.93e3·9-s + 1.51e4·10-s + 2.33e3·11-s − 4.90e4·12-s + 3.41e4·13-s − 6.01e4·14-s − 1.76e5·15-s + 1.07e5·16-s + 5.13e4·18-s − 2.47e5·19-s + 5.88e5·20-s + 7.02e5·21-s − 2.42e4·22-s − 6.66e5·23-s + 1.15e6·24-s + 1.64e5·25-s − 3.54e5·26-s − 2.98e6·27-s − 2.33e6·28-s − 2.98e6·29-s + ⋯
L(s)  = 1  − 0.459·2-s + 0.865·3-s − 0.789·4-s − 1.04·5-s − 0.397·6-s + 0.911·7-s + 0.821·8-s − 0.250·9-s + 0.478·10-s + 0.0480·11-s − 0.683·12-s + 0.331·13-s − 0.418·14-s − 0.901·15-s + 0.411·16-s + 0.115·18-s − 0.434·19-s + 0.821·20-s + 0.788·21-s − 0.0220·22-s − 0.496·23-s + 0.711·24-s + 0.0844·25-s − 0.152·26-s − 1.08·27-s − 0.718·28-s − 0.784·29-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 \( 1 + 10.3T + 512T^{2} \)
3 \( 1 - 121.T + 1.96e4T^{2} \)
5 \( 1 + 1.45e3T + 1.95e6T^{2} \)
7 \( 1 - 5.78e3T + 4.03e7T^{2} \)
11 \( 1 - 2.33e3T + 2.35e9T^{2} \)
13 \( 1 - 3.41e4T + 1.06e10T^{2} \)
19 \( 1 + 2.47e5T + 3.22e11T^{2} \)
23 \( 1 + 6.66e5T + 1.80e12T^{2} \)
29 \( 1 + 2.98e6T + 1.45e13T^{2} \)
31 \( 1 - 7.53e6T + 2.64e13T^{2} \)
37 \( 1 + 9.81e5T + 1.29e14T^{2} \)
41 \( 1 - 2.32e7T + 3.27e14T^{2} \)
43 \( 1 - 2.36e7T + 5.02e14T^{2} \)
47 \( 1 - 5.99e7T + 1.11e15T^{2} \)
53 \( 1 - 5.88e7T + 3.29e15T^{2} \)
59 \( 1 + 1.19e8T + 8.66e15T^{2} \)
61 \( 1 - 1.73e8T + 1.16e16T^{2} \)
67 \( 1 + 2.80e8T + 2.72e16T^{2} \)
71 \( 1 - 1.98e7T + 4.58e16T^{2} \)
73 \( 1 - 3.38e8T + 5.88e16T^{2} \)
79 \( 1 - 1.90e8T + 1.19e17T^{2} \)
83 \( 1 + 5.33e8T + 1.86e17T^{2} \)
89 \( 1 + 7.82e8T + 3.50e17T^{2} \)
97 \( 1 + 1.10e9T + 7.60e17T^{2} \)
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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

−9.447439095865377338222789774242, −8.618554028073938180909139077940, −8.066362825524370172701421722614, −7.46362243981533741259283750933, −5.70024830538364662657271107413, −4.37645252314764598143856361140, −3.83011575619542482826453421914, −2.44157248097370132900946520195, −1.10740188555563700749446435955, 0, 1.10740188555563700749446435955, 2.44157248097370132900946520195, 3.83011575619542482826453421914, 4.37645252314764598143856361140, 5.70024830538364662657271107413, 7.46362243981533741259283750933, 8.066362825524370172701421722614, 8.618554028073938180909139077940, 9.447439095865377338222789774242

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