Properties

Degree $2$
Conductor $17$
Sign $1$
Motivic weight $9$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.12·2-s − 254.·3-s − 494.·4-s + 151.·5-s − 1.04e3·6-s + 9.40e3·7-s − 4.15e3·8-s + 4.48e4·9-s + 625.·10-s − 5.69e4·11-s + 1.25e5·12-s − 6.08e4·13-s + 3.88e4·14-s − 3.85e4·15-s + 2.36e5·16-s + 8.35e4·17-s + 1.85e5·18-s + 1.00e6·19-s − 7.50e4·20-s − 2.39e6·21-s − 2.35e5·22-s + 1.35e6·23-s + 1.05e6·24-s − 1.93e6·25-s − 2.51e5·26-s − 6.39e6·27-s − 4.65e6·28-s + ⋯
L(s)  = 1  + 0.182·2-s − 1.81·3-s − 0.966·4-s + 0.108·5-s − 0.330·6-s + 1.48·7-s − 0.358·8-s + 2.27·9-s + 0.0197·10-s − 1.17·11-s + 1.75·12-s − 0.591·13-s + 0.270·14-s − 0.196·15-s + 0.901·16-s + 0.242·17-s + 0.416·18-s + 1.77·19-s − 0.104·20-s − 2.68·21-s − 0.214·22-s + 1.01·23-s + 0.650·24-s − 0.988·25-s − 0.107·26-s − 2.31·27-s − 1.43·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $1$
Motivic weight: \(9\)
Character: $\chi_{17} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.842276\)
\(L(\frac12)\) \(\approx\) \(0.842276\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - 8.35e4T \)
good2 \( 1 - 4.12T + 512T^{2} \)
3 \( 1 + 254.T + 1.96e4T^{2} \)
5 \( 1 - 151.T + 1.95e6T^{2} \)
7 \( 1 - 9.40e3T + 4.03e7T^{2} \)
11 \( 1 + 5.69e4T + 2.35e9T^{2} \)
13 \( 1 + 6.08e4T + 1.06e10T^{2} \)
19 \( 1 - 1.00e6T + 3.22e11T^{2} \)
23 \( 1 - 1.35e6T + 1.80e12T^{2} \)
29 \( 1 - 3.12e6T + 1.45e13T^{2} \)
31 \( 1 - 2.97e6T + 2.64e13T^{2} \)
37 \( 1 - 6.81e5T + 1.29e14T^{2} \)
41 \( 1 + 4.09e6T + 3.27e14T^{2} \)
43 \( 1 - 1.00e7T + 5.02e14T^{2} \)
47 \( 1 - 2.54e7T + 1.11e15T^{2} \)
53 \( 1 + 3.14e7T + 3.29e15T^{2} \)
59 \( 1 + 9.03e7T + 8.66e15T^{2} \)
61 \( 1 - 9.87e7T + 1.16e16T^{2} \)
67 \( 1 - 1.32e8T + 2.72e16T^{2} \)
71 \( 1 - 4.18e8T + 4.58e16T^{2} \)
73 \( 1 - 4.80e7T + 5.88e16T^{2} \)
79 \( 1 - 3.49e7T + 1.19e17T^{2} \)
83 \( 1 + 2.05e8T + 1.86e17T^{2} \)
89 \( 1 + 2.03e8T + 3.50e17T^{2} \)
97 \( 1 - 1.24e9T + 7.60e17T^{2} \)
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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

−17.17096707864571088822781875455, −15.58059279670676830910762326962, −13.88180122252500421742811551628, −12.40966610854059651971919210545, −11.30396254096535651521154351679, −9.987133772954341697833874929165, −7.70653694709031935913070747997, −5.41543927693687106766573064045, −4.82801047484950600224407818281, −0.849171790121337493242783066568, 0.849171790121337493242783066568, 4.82801047484950600224407818281, 5.41543927693687106766573064045, 7.70653694709031935913070747997, 9.987133772954341697833874929165, 11.30396254096535651521154351679, 12.40966610854059651971919210545, 13.88180122252500421742811551628, 15.58059279670676830910762326962, 17.17096707864571088822781875455

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