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Properties

Label 2-17e2-17.16-c3-0-45
Degree $2$
Conductor $289$
Sign $-0.743 + 0.669i$
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  − 4.28·2-s + 2.85i·3-s + 10.3·4-s − 6.36i·5-s − 12.2i·6-s − 29.4i·7-s − 10.2·8-s + 18.8·9-s + 27.3i·10-s − 61.3i·11-s + 29.6i·12-s + 18.5·13-s + 126. i·14-s + 18.1·15-s − 39.1·16-s + ⋯
L(s)  = 1  − 1.51·2-s + 0.548i·3-s + 1.29·4-s − 0.569i·5-s − 0.831i·6-s − 1.58i·7-s − 0.453·8-s + 0.699·9-s + 0.863i·10-s − 1.68i·11-s + 0.712i·12-s + 0.395·13-s + 2.40i·14-s + 0.312·15-s − 0.611·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.743 + 0.669i$
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: \((2,\ 289,\ (\ :3/2),\ -0.743 + 0.669i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5875495455\)
\(L(\frac12)\) \(\approx\) \(0.5875495455\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.28T + 8T^{2} \)
3 \( 1 - 2.85iT - 27T^{2} \)
5 \( 1 + 6.36iT - 125T^{2} \)
7 \( 1 + 29.4iT - 343T^{2} \)
11 \( 1 + 61.3iT - 1.33e3T^{2} \)
13 \( 1 - 18.5T + 2.19e3T^{2} \)
19 \( 1 + 115.T + 6.85e3T^{2} \)
23 \( 1 - 7.38iT - 1.21e4T^{2} \)
29 \( 1 - 164. iT - 2.43e4T^{2} \)
31 \( 1 + 127. iT - 2.97e4T^{2} \)
37 \( 1 + 158. iT - 5.06e4T^{2} \)
41 \( 1 - 31.3iT - 6.89e4T^{2} \)
43 \( 1 + 157.T + 7.95e4T^{2} \)
47 \( 1 + 460.T + 1.03e5T^{2} \)
53 \( 1 - 166.T + 1.48e5T^{2} \)
59 \( 1 - 343.T + 2.05e5T^{2} \)
61 \( 1 + 112. iT - 2.26e5T^{2} \)
67 \( 1 + 984.T + 3.00e5T^{2} \)
71 \( 1 - 524. iT - 3.57e5T^{2} \)
73 \( 1 + 852. iT - 3.89e5T^{2} \)
79 \( 1 + 201. iT - 4.93e5T^{2} \)
83 \( 1 - 22.5T + 5.71e5T^{2} \)
89 \( 1 - 502.T + 7.04e5T^{2} \)
97 \( 1 + 680. iT - 9.12e5T^{2} \)
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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

−10.68981061326581586231440335587, −10.14577126258411283756696405826, −9.021518029633155539689343089514, −8.405351881331427309490982939408, −7.40226628067198032876990901463, −6.41248441336270205285311890597, −4.66598292163326709189945581641, −3.62080143464918254354128672601, −1.36014464833314687185882533811, −0.39765925977043359530612457843, 1.68083708856015830397038680867, 2.43291134722682744778379398346, 4.64233985225450683141899688347, 6.37829049555815908603153747535, 6.99872222267233978234692608597, 8.056120610585986127587364639683, 8.845088586584952638188431898710, 9.801033812479823998634505346345, 10.44178724256916062065940106565, 11.62959219296678053165285069646

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