Properties

Label 2-17e2-1.1-c9-0-75
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  − 13.2·2-s − 162.·3-s − 337.·4-s − 1.69e3·5-s + 2.15e3·6-s + 1.04e4·7-s + 1.12e4·8-s + 6.74e3·9-s + 2.23e4·10-s − 2.79e4·11-s + 5.47e4·12-s − 1.47e5·13-s − 1.37e5·14-s + 2.74e5·15-s + 2.40e4·16-s − 8.92e4·18-s − 5.05e5·19-s + 5.69e5·20-s − 1.69e6·21-s + 3.69e5·22-s − 1.59e6·23-s − 1.82e6·24-s + 9.06e5·25-s + 1.95e6·26-s + 2.10e6·27-s − 3.51e6·28-s − 3.23e6·29-s + ⋯
L(s)  = 1  − 0.584·2-s − 1.15·3-s − 0.658·4-s − 1.20·5-s + 0.677·6-s + 1.64·7-s + 0.969·8-s + 0.342·9-s + 0.707·10-s − 0.574·11-s + 0.762·12-s − 1.43·13-s − 0.959·14-s + 1.40·15-s + 0.0915·16-s − 0.200·18-s − 0.890·19-s + 0.796·20-s − 1.90·21-s + 0.335·22-s − 1.18·23-s − 1.12·24-s + 0.463·25-s + 0.837·26-s + 0.761·27-s − 1.08·28-s − 0.849·29-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: Trivial
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 \( 1 + 13.2T + 512T^{2} \)
3 \( 1 + 162.T + 1.96e4T^{2} \)
5 \( 1 + 1.69e3T + 1.95e6T^{2} \)
7 \( 1 - 1.04e4T + 4.03e7T^{2} \)
11 \( 1 + 2.79e4T + 2.35e9T^{2} \)
13 \( 1 + 1.47e5T + 1.06e10T^{2} \)
19 \( 1 + 5.05e5T + 3.22e11T^{2} \)
23 \( 1 + 1.59e6T + 1.80e12T^{2} \)
29 \( 1 + 3.23e6T + 1.45e13T^{2} \)
31 \( 1 - 5.16e6T + 2.64e13T^{2} \)
37 \( 1 - 1.45e7T + 1.29e14T^{2} \)
41 \( 1 + 3.32e7T + 3.27e14T^{2} \)
43 \( 1 - 2.69e6T + 5.02e14T^{2} \)
47 \( 1 - 9.23e6T + 1.11e15T^{2} \)
53 \( 1 - 6.19e7T + 3.29e15T^{2} \)
59 \( 1 - 9.85e7T + 8.66e15T^{2} \)
61 \( 1 - 2.07e8T + 1.16e16T^{2} \)
67 \( 1 - 5.69e7T + 2.72e16T^{2} \)
71 \( 1 + 1.84e8T + 4.58e16T^{2} \)
73 \( 1 + 3.74e8T + 5.88e16T^{2} \)
79 \( 1 - 1.08e8T + 1.19e17T^{2} \)
83 \( 1 - 4.94e8T + 1.86e17T^{2} \)
89 \( 1 + 1.69e8T + 3.50e17T^{2} \)
97 \( 1 - 7.68e8T + 7.60e17T^{2} \)
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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

−10.00613201393215790264928471905, −8.460351824069947243961719069924, −8.007687376601129591216252470703, −7.17906989622370291773047520294, −5.51355698825878247698871476412, −4.75068316923247904322639725617, −4.15390101913796369041619645403, −2.11100414589467856448994549255, −0.70617919101925379681968462268, 0, 0.70617919101925379681968462268, 2.11100414589467856448994549255, 4.15390101913796369041619645403, 4.75068316923247904322639725617, 5.51355698825878247698871476412, 7.17906989622370291773047520294, 8.007687376601129591216252470703, 8.460351824069947243961719069924, 10.00613201393215790264928471905

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