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Properties

Label 2-17e2-17.16-c3-0-34
Degree $2$
Conductor $289$
Sign $-0.242 - 0.970i$
Analytic cond. $17.0515$
Root an. cond. $4.12935$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.03·2-s + 8.47i·3-s + 17.3·4-s + 0.885i·5-s + 42.6i·6-s − 3.81i·7-s + 46.9·8-s − 44.8·9-s + 4.45i·10-s + 52.3i·11-s + 146. i·12-s − 8.06·13-s − 19.2i·14-s − 7.50·15-s + 97.5·16-s + ⋯
L(s)  = 1  + 1.77·2-s + 1.63i·3-s + 2.16·4-s + 0.0792i·5-s + 2.90i·6-s − 0.206i·7-s + 2.07·8-s − 1.66·9-s + 0.140i·10-s + 1.43i·11-s + 3.53i·12-s − 0.171·13-s − 0.366i·14-s − 0.129·15-s + 1.52·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(17.0515\)
Root analytic conductor: \(4.12935\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ -0.242 - 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(5.129981105\)
\(L(\frac12)\) \(\approx\) \(5.129981105\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.03T + 8T^{2} \)
3 \( 1 - 8.47iT - 27T^{2} \)
5 \( 1 - 0.885iT - 125T^{2} \)
7 \( 1 + 3.81iT - 343T^{2} \)
11 \( 1 - 52.3iT - 1.33e3T^{2} \)
13 \( 1 + 8.06T + 2.19e3T^{2} \)
19 \( 1 - 66.5T + 6.85e3T^{2} \)
23 \( 1 + 180. iT - 1.21e4T^{2} \)
29 \( 1 + 41.2iT - 2.43e4T^{2} \)
31 \( 1 + 34.9iT - 2.97e4T^{2} \)
37 \( 1 - 130. iT - 5.06e4T^{2} \)
41 \( 1 - 17.9iT - 6.89e4T^{2} \)
43 \( 1 + 277.T + 7.95e4T^{2} \)
47 \( 1 - 463.T + 1.03e5T^{2} \)
53 \( 1 - 329.T + 1.48e5T^{2} \)
59 \( 1 + 678.T + 2.05e5T^{2} \)
61 \( 1 + 340. iT - 2.26e5T^{2} \)
67 \( 1 - 15.3T + 3.00e5T^{2} \)
71 \( 1 + 670. iT - 3.57e5T^{2} \)
73 \( 1 - 193. iT - 3.89e5T^{2} \)
79 \( 1 + 1.08e3iT - 4.93e5T^{2} \)
83 \( 1 - 865.T + 5.71e5T^{2} \)
89 \( 1 - 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 379. iT - 9.12e5T^{2} \)
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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

−11.84203694718048272013254082877, −10.71037754337493683011477177857, −10.18730726042960785301133254942, −9.053533761779657735156711548187, −7.38957205089268302469944248614, −6.28265526833325049177418430553, −4.95835387835224406851643669154, −4.62965925728495872795447302215, −3.62218137517693172567155662428, −2.50478052096995127948117356407, 1.17520300580676906113474682332, 2.58679547992551004433056186428, 3.54578393583029545154327722618, 5.32623561913139519928201347962, 5.90613144343931022078424376403, 6.91282416417120081673763084003, 7.65908850200305231374285658564, 8.883338491254753127958586834490, 10.85070209073163852156106669885, 11.69133750323761602781401384128

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