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Properties

Label 2-17-17.16-c9-0-7
Degree $2$
Conductor $17$
Sign $0.839 - 0.543i$
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  + 39.2·2-s + 12.8i·3-s + 1.02e3·4-s + 2.41e3i·5-s + 503. i·6-s − 2.30e3i·7-s + 2.02e4·8-s + 1.95e4·9-s + 9.47e4i·10-s − 8.14e4i·11-s + 1.31e4i·12-s − 5.93e4·13-s − 9.03e4i·14-s − 3.09e4·15-s + 2.68e5·16-s + (−2.89e5 + 1.87e5i)17-s + ⋯
L(s)  = 1  + 1.73·2-s + 0.0915i·3-s + 2.00·4-s + 1.72i·5-s + 0.158i·6-s − 0.362i·7-s + 1.74·8-s + 0.991·9-s + 2.99i·10-s − 1.67i·11-s + 0.183i·12-s − 0.576·13-s − 0.628i·14-s − 0.158·15-s + 1.02·16-s + (−0.839 + 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.839 - 0.543i$
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: \((2,\ 17,\ (\ :9/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(5)\) \(\approx\) \(4.20320 + 1.24243i\)
\(L(\frac12)\) \(\approx\) \(4.20320 + 1.24243i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.89e5 - 1.87e5i)T \)
good2 \( 1 - 39.2T + 512T^{2} \)
3 \( 1 - 12.8iT - 1.96e4T^{2} \)
5 \( 1 - 2.41e3iT - 1.95e6T^{2} \)
7 \( 1 + 2.30e3iT - 4.03e7T^{2} \)
11 \( 1 + 8.14e4iT - 2.35e9T^{2} \)
13 \( 1 + 5.93e4T + 1.06e10T^{2} \)
19 \( 1 - 3.66e5T + 3.22e11T^{2} \)
23 \( 1 + 1.24e6iT - 1.80e12T^{2} \)
29 \( 1 - 1.35e6iT - 1.45e13T^{2} \)
31 \( 1 + 4.78e6iT - 2.64e13T^{2} \)
37 \( 1 + 8.36e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.02e7iT - 3.27e14T^{2} \)
43 \( 1 - 2.33e7T + 5.02e14T^{2} \)
47 \( 1 + 3.90e7T + 1.11e15T^{2} \)
53 \( 1 - 2.82e7T + 3.29e15T^{2} \)
59 \( 1 - 2.09e7T + 8.66e15T^{2} \)
61 \( 1 - 1.68e8iT - 1.16e16T^{2} \)
67 \( 1 + 1.42e8T + 2.72e16T^{2} \)
71 \( 1 - 3.49e8iT - 4.58e16T^{2} \)
73 \( 1 + 2.50e8iT - 5.88e16T^{2} \)
79 \( 1 - 1.12e8iT - 1.19e17T^{2} \)
83 \( 1 + 5.17e8T + 1.86e17T^{2} \)
89 \( 1 + 3.46e7T + 3.50e17T^{2} \)
97 \( 1 + 7.57e8iT - 7.60e17T^{2} \)
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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

−16.22277087366625083124567774201, −15.05558814328202019493996722717, −14.16791976926534245683124472630, −13.16343242665955402336126319310, −11.42076822807046790791682388026, −10.51276058318630736938637816579, −7.19168898322805231070951159806, −6.08972515061319450906711888591, −3.99939777622717558710339755252, −2.72624736277103472784973707427, 1.83091115279980781942673744034, 4.39439223587545790291293910052, 5.15143163604946407389243489859, 7.22026508346836077068609258340, 9.519397495315872494216591735420, 12.02764314515844533989069111186, 12.58982101654063360194529498157, 13.56203962492967231169901651906, 15.29916329700073775945807218595, 15.99125871188734003631075636464

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