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Properties

Label 2-17e2-17.16-c3-0-40
Degree $2$
Conductor $289$
Sign $-0.168 + 0.985i$
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  − 3.25·2-s + 6.51i·3-s + 2.61·4-s − 19.1i·5-s − 21.2i·6-s + 22.0i·7-s + 17.5·8-s − 15.4·9-s + 62.2i·10-s − 8.25i·11-s + 17.0i·12-s − 0.398·13-s − 71.9i·14-s + 124.·15-s − 78.0·16-s + ⋯
L(s)  = 1  − 1.15·2-s + 1.25i·3-s + 0.326·4-s − 1.70i·5-s − 1.44i·6-s + 1.19i·7-s + 0.775·8-s − 0.573·9-s + 1.96i·10-s − 0.226i·11-s + 0.409i·12-s − 0.00849·13-s − 1.37i·14-s + 2.14·15-s − 1.21·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.168 + 0.985i$
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: \((2,\ 289,\ (\ :3/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2991133421\)
\(L(\frac12)\) \(\approx\) \(0.2991133421\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.25T + 8T^{2} \)
3 \( 1 - 6.51iT - 27T^{2} \)
5 \( 1 + 19.1iT - 125T^{2} \)
7 \( 1 - 22.0iT - 343T^{2} \)
11 \( 1 + 8.25iT - 1.33e3T^{2} \)
13 \( 1 + 0.398T + 2.19e3T^{2} \)
19 \( 1 + 103.T + 6.85e3T^{2} \)
23 \( 1 - 138. iT - 1.21e4T^{2} \)
29 \( 1 + 222. iT - 2.43e4T^{2} \)
31 \( 1 + 49.1iT - 2.97e4T^{2} \)
37 \( 1 + 61.3iT - 5.06e4T^{2} \)
41 \( 1 + 387. iT - 6.89e4T^{2} \)
43 \( 1 - 20.3T + 7.95e4T^{2} \)
47 \( 1 - 44.1T + 1.03e5T^{2} \)
53 \( 1 + 59.6T + 1.48e5T^{2} \)
59 \( 1 + 238.T + 2.05e5T^{2} \)
61 \( 1 + 595. iT - 2.26e5T^{2} \)
67 \( 1 + 408.T + 3.00e5T^{2} \)
71 \( 1 + 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 + 22.3iT - 3.89e5T^{2} \)
79 \( 1 + 682. iT - 4.93e5T^{2} \)
83 \( 1 - 312.T + 5.71e5T^{2} \)
89 \( 1 + 904.T + 7.04e5T^{2} \)
97 \( 1 + 1.00e3iT - 9.12e5T^{2} \)
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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

−10.79638059410683144727572756718, −9.733366044554101553146468317487, −9.169932991610179037738322614328, −8.709024911207428971693850554443, −7.84686349270508162515093690593, −5.80838886633044234008583104882, −4.90368809973910944862347788086, −4.04455653403196002720007266075, −1.85639703527155227560712308490, −0.18043293960027995470574908834, 1.29175655816415088087778027091, 2.58950113186042929897090702299, 4.22931219224325158567900564780, 6.51475647818315846663127890048, 6.93876210058112637322409971507, 7.61566771156754476647117407926, 8.510583312624459677210943875971, 10.00978815102098834841958775674, 10.56711475648889719790961841045, 11.19388047692218341584540441190

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