Properties

Label 2-17e2-1.1-c9-0-78
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  − 43.7·2-s − 83.5·3-s + 1.40e3·4-s + 712.·5-s + 3.65e3·6-s − 5.54e3·7-s − 3.89e4·8-s − 1.26e4·9-s − 3.11e4·10-s − 3.60e4·11-s − 1.17e5·12-s − 1.75e5·13-s + 2.42e5·14-s − 5.95e4·15-s + 9.84e5·16-s + 5.55e5·18-s − 5.12e5·19-s + 9.98e5·20-s + 4.63e5·21-s + 1.57e6·22-s + 1.41e6·23-s + 3.25e6·24-s − 1.44e6·25-s + 7.67e6·26-s + 2.70e6·27-s − 7.76e6·28-s + 6.23e6·29-s + ⋯
L(s)  = 1  − 1.93·2-s − 0.595·3-s + 2.73·4-s + 0.509·5-s + 1.15·6-s − 0.872·7-s − 3.35·8-s − 0.644·9-s − 0.985·10-s − 0.743·11-s − 1.63·12-s − 1.70·13-s + 1.68·14-s − 0.303·15-s + 3.75·16-s + 1.24·18-s − 0.902·19-s + 1.39·20-s + 0.519·21-s + 1.43·22-s + 1.05·23-s + 2.00·24-s − 0.740·25-s + 3.29·26-s + 0.980·27-s − 2.38·28-s + 1.63·29-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 \( 1 + 43.7T + 512T^{2} \)
3 \( 1 + 83.5T + 1.96e4T^{2} \)
5 \( 1 - 712.T + 1.95e6T^{2} \)
7 \( 1 + 5.54e3T + 4.03e7T^{2} \)
11 \( 1 + 3.60e4T + 2.35e9T^{2} \)
13 \( 1 + 1.75e5T + 1.06e10T^{2} \)
19 \( 1 + 5.12e5T + 3.22e11T^{2} \)
23 \( 1 - 1.41e6T + 1.80e12T^{2} \)
29 \( 1 - 6.23e6T + 1.45e13T^{2} \)
31 \( 1 - 1.00e5T + 2.64e13T^{2} \)
37 \( 1 - 1.77e7T + 1.29e14T^{2} \)
41 \( 1 + 2.70e7T + 3.27e14T^{2} \)
43 \( 1 - 7.78e5T + 5.02e14T^{2} \)
47 \( 1 - 1.03e7T + 1.11e15T^{2} \)
53 \( 1 - 3.35e7T + 3.29e15T^{2} \)
59 \( 1 - 1.32e8T + 8.66e15T^{2} \)
61 \( 1 - 4.00e7T + 1.16e16T^{2} \)
67 \( 1 - 2.73e8T + 2.72e16T^{2} \)
71 \( 1 + 1.50e8T + 4.58e16T^{2} \)
73 \( 1 + 1.55e7T + 5.88e16T^{2} \)
79 \( 1 - 4.95e7T + 1.19e17T^{2} \)
83 \( 1 + 2.23e8T + 1.86e17T^{2} \)
89 \( 1 - 3.53e7T + 3.50e17T^{2} \)
97 \( 1 - 1.41e9T + 7.60e17T^{2} \)
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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

−9.960593838824474536379847879837, −8.926846419625656182062959434182, −8.028379750971990761718536638914, −6.94181220028482448644306614783, −6.29064451850218595286568807680, −5.20806478353818132942264414599, −2.83757287330653248847249689010, −2.29472130319417651814305138926, −0.72308549512638738047794863888, 0, 0.72308549512638738047794863888, 2.29472130319417651814305138926, 2.83757287330653248847249689010, 5.20806478353818132942264414599, 6.29064451850218595286568807680, 6.94181220028482448644306614783, 8.028379750971990761718536638914, 8.926846419625656182062959434182, 9.960593838824474536379847879837

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