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Properties

Label 2-17e2-17.16-c3-0-31
Degree $2$
Conductor $289$
Sign $0.911 - 0.410i$
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  − 4.44·2-s + 0.537i·3-s + 11.7·4-s + 17.2i·5-s − 2.38i·6-s + 6.40i·7-s − 16.6·8-s + 26.7·9-s − 76.6i·10-s − 55.3i·11-s + 6.31i·12-s + 58.6·13-s − 28.4i·14-s − 9.27·15-s − 20.0·16-s + ⋯
L(s)  = 1  − 1.57·2-s + 0.103i·3-s + 1.46·4-s + 1.54i·5-s − 0.162i·6-s + 0.345i·7-s − 0.735·8-s + 0.989·9-s − 2.42i·10-s − 1.51i·11-s + 0.151i·12-s + 1.25·13-s − 0.543i·14-s − 0.159·15-s − 0.312·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.911 - 0.410i$
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: \((2,\ 289,\ (\ :3/2),\ 0.911 - 0.410i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9751102261\)
\(L(\frac12)\) \(\approx\) \(0.9751102261\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.44T + 8T^{2} \)
3 \( 1 - 0.537iT - 27T^{2} \)
5 \( 1 - 17.2iT - 125T^{2} \)
7 \( 1 - 6.40iT - 343T^{2} \)
11 \( 1 + 55.3iT - 1.33e3T^{2} \)
13 \( 1 - 58.6T + 2.19e3T^{2} \)
19 \( 1 - 91.1T + 6.85e3T^{2} \)
23 \( 1 + 120. iT - 1.21e4T^{2} \)
29 \( 1 + 215. iT - 2.43e4T^{2} \)
31 \( 1 + 17.5iT - 2.97e4T^{2} \)
37 \( 1 + 8.40iT - 5.06e4T^{2} \)
41 \( 1 + 99.9iT - 6.89e4T^{2} \)
43 \( 1 - 81.5T + 7.95e4T^{2} \)
47 \( 1 - 195.T + 1.03e5T^{2} \)
53 \( 1 + 260.T + 1.48e5T^{2} \)
59 \( 1 - 536.T + 2.05e5T^{2} \)
61 \( 1 - 265. iT - 2.26e5T^{2} \)
67 \( 1 - 514.T + 3.00e5T^{2} \)
71 \( 1 + 704. iT - 3.57e5T^{2} \)
73 \( 1 - 184. iT - 3.89e5T^{2} \)
79 \( 1 - 34.8iT - 4.93e5T^{2} \)
83 \( 1 - 647.T + 5.71e5T^{2} \)
89 \( 1 + 1.06e3T + 7.04e5T^{2} \)
97 \( 1 + 256. iT - 9.12e5T^{2} \)
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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

−10.96665706704717216883580580382, −10.50988454499913318508908451096, −9.601091655914351922258522441995, −8.598010815016593652213096767471, −7.73210805679398814214208084961, −6.76508389272588424712484542332, −5.97745650470901974709104847286, −3.74250957017171540634028827513, −2.50183767536605350856376448429, −0.866075371627149751959561819160, 1.06100393681866511906607937295, 1.60062764484713216156188803845, 4.09291806072272521031163731337, 5.21395862892281612910098410743, 6.93757305233633394017103099647, 7.63101013991642588587598434539, 8.586364135818488941126237316193, 9.446294215711931093371775656563, 9.922323664994409702829441600161, 11.04472838884283867548309788349

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