Properties

Label 2-17e2-17.16-c3-0-55
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  + 4.98·2-s − 6.26i·3-s + 16.8·4-s − 15.7i·5-s − 31.1i·6-s + 0.789i·7-s + 43.8·8-s − 12.2·9-s − 78.4i·10-s + 45.3i·11-s − 105. i·12-s + 46.7·13-s + 3.93i·14-s − 98.6·15-s + 84.1·16-s + ⋯
L(s)  = 1  + 1.76·2-s − 1.20i·3-s + 2.10·4-s − 1.40i·5-s − 2.12i·6-s + 0.0426i·7-s + 1.93·8-s − 0.452·9-s − 2.48i·10-s + 1.24i·11-s − 2.53i·12-s + 0.997·13-s + 0.0751i·14-s − 1.69·15-s + 1.31·16-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

\(L(2)\) \(\approx\) \(5.272205116\)
\(L(\frac12)\) \(\approx\) \(5.272205116\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.98T + 8T^{2} \)
3 \( 1 + 6.26iT - 27T^{2} \)
5 \( 1 + 15.7iT - 125T^{2} \)
7 \( 1 - 0.789iT - 343T^{2} \)
11 \( 1 - 45.3iT - 1.33e3T^{2} \)
13 \( 1 - 46.7T + 2.19e3T^{2} \)
19 \( 1 + 100.T + 6.85e3T^{2} \)
23 \( 1 + 84.1iT - 1.21e4T^{2} \)
29 \( 1 - 101. iT - 2.43e4T^{2} \)
31 \( 1 + 7.36iT - 2.97e4T^{2} \)
37 \( 1 - 251. iT - 5.06e4T^{2} \)
41 \( 1 - 260. iT - 6.89e4T^{2} \)
43 \( 1 - 401.T + 7.95e4T^{2} \)
47 \( 1 - 304.T + 1.03e5T^{2} \)
53 \( 1 + 398.T + 1.48e5T^{2} \)
59 \( 1 - 577.T + 2.05e5T^{2} \)
61 \( 1 - 126. iT - 2.26e5T^{2} \)
67 \( 1 + 150.T + 3.00e5T^{2} \)
71 \( 1 + 434. iT - 3.57e5T^{2} \)
73 \( 1 + 493. iT - 3.89e5T^{2} \)
79 \( 1 + 72.2iT - 4.93e5T^{2} \)
83 \( 1 + 711.T + 5.71e5T^{2} \)
89 \( 1 - 1.35e3T + 7.04e5T^{2} \)
97 \( 1 - 1.40e3iT - 9.12e5T^{2} \)
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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

−11.85778534529216986028602669792, −10.58626284151782150712580350408, −8.954954126132961850085602491541, −7.891338122682839357542505688047, −6.79118623708185038619141652119, −6.03919989943313974894556990546, −4.83747341570675182554331122599, −4.13253088518212597321825024083, −2.31709296793214027740872171486, −1.26280709648573248284289355574, 2.55475810481551180532941545873, 3.64761336082154479816490529361, 4.04769920145317820419438683074, 5.58461809481282927766587002741, 6.19650587579216354658422566531, 7.28473969773933001288852488212, 8.863208102867133343035503391654, 10.34871390085966874807276845544, 10.97979694700039700528758303656, 11.33665514474235702901249879348

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