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Properties

Label 2-17-17.16-c9-0-0
Degree $2$
Conductor $17$
Sign $-0.740 + 0.671i$
Analytic cond. $8.75560$
Root an. cond. $2.95898$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.7·2-s + 206. i·3-s − 231.·4-s + 1.94e3i·5-s − 3.46e3i·6-s + 1.63e3i·7-s + 1.24e4·8-s − 2.30e4·9-s − 3.26e4i·10-s − 1.16e4i·11-s − 4.78e4i·12-s − 1.63e5·13-s − 2.73e4i·14-s − 4.02e5·15-s − 9.01e4·16-s + (2.55e5 − 2.31e5i)17-s + ⋯
L(s)  = 1  − 0.740·2-s + 1.47i·3-s − 0.451·4-s + 1.39i·5-s − 1.09i·6-s + 0.257i·7-s + 1.07·8-s − 1.16·9-s − 1.03i·10-s − 0.239i·11-s − 0.665i·12-s − 1.58·13-s − 0.190i·14-s − 2.05·15-s − 0.344·16-s + (0.740 − 0.671i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.740 + 0.671i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.740 + 0.671i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.204919 - 0.531256i\)
\(L(\frac12)\) \(\approx\) \(0.204919 - 0.531256i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.55e5 + 2.31e5i)T \)
good2 \( 1 + 16.7T + 512T^{2} \)
3 \( 1 - 206. iT - 1.96e4T^{2} \)
5 \( 1 - 1.94e3iT - 1.95e6T^{2} \)
7 \( 1 - 1.63e3iT - 4.03e7T^{2} \)
11 \( 1 + 1.16e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.63e5T + 1.06e10T^{2} \)
19 \( 1 - 6.73e5T + 3.22e11T^{2} \)
23 \( 1 - 1.37e6iT - 1.80e12T^{2} \)
29 \( 1 + 4.32e6iT - 1.45e13T^{2} \)
31 \( 1 - 3.45e6iT - 2.64e13T^{2} \)
37 \( 1 + 5.81e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.37e7iT - 3.27e14T^{2} \)
43 \( 1 + 2.46e7T + 5.02e14T^{2} \)
47 \( 1 + 3.94e7T + 1.11e15T^{2} \)
53 \( 1 + 9.66e7T + 3.29e15T^{2} \)
59 \( 1 - 2.37e7T + 8.66e15T^{2} \)
61 \( 1 - 4.31e7iT - 1.16e16T^{2} \)
67 \( 1 - 3.50e7T + 2.72e16T^{2} \)
71 \( 1 + 2.90e8iT - 4.58e16T^{2} \)
73 \( 1 - 4.29e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.78e8iT - 1.19e17T^{2} \)
83 \( 1 - 9.91e7T + 1.86e17T^{2} \)
89 \( 1 + 5.39e8T + 3.50e17T^{2} \)
97 \( 1 - 1.21e9iT - 7.60e17T^{2} \)
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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

−17.55320266858113284434537853500, −16.28808596043195410775156086760, −14.98670838726775677927529653671, −14.05985783751146915148032648240, −11.44828480690860949195397268021, −10.04200651141327650887764278687, −9.583169333801932846768357577097, −7.55006900803666144679868607794, −5.04571119783555791788227382117, −3.19857669712475036449023535277, 0.38940199225505379615176512350, 1.52124544103895810372268717156, 4.99823270401270205327722504749, 7.35520418486498187997395021916, 8.378944863839360814930988065079, 9.794155700545873277183867792751, 12.23196424154973388335382233447, 12.90979339724986848399986349087, 14.18871477195835535321394123328, 16.65442271806713555372821950129

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