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Properties

Label 2-17e2-17.16-c3-0-13
Degree $2$
Conductor $289$
Sign $-0.970 - 0.242i$
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  + 3.68·2-s + 9.05i·3-s + 5.57·4-s − 7.08i·5-s + 33.3i·6-s + 28.1i·7-s − 8.92·8-s − 55.0·9-s − 26.1i·10-s − 15.3i·11-s + 50.5i·12-s + 2.51·13-s + 103. i·14-s + 64.2·15-s − 77.5·16-s + ⋯
L(s)  = 1  + 1.30·2-s + 1.74i·3-s + 0.697·4-s − 0.634i·5-s + 2.27i·6-s + 1.52i·7-s − 0.394·8-s − 2.03·9-s − 0.826i·10-s − 0.419i·11-s + 1.21i·12-s + 0.0536·13-s + 1.98i·14-s + 1.10·15-s − 1.21·16-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.970 - 0.242i$
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: \((2,\ 289,\ (\ :3/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.521007359\)
\(L(\frac12)\) \(\approx\) \(2.521007359\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.68T + 8T^{2} \)
3 \( 1 - 9.05iT - 27T^{2} \)
5 \( 1 + 7.08iT - 125T^{2} \)
7 \( 1 - 28.1iT - 343T^{2} \)
11 \( 1 + 15.3iT - 1.33e3T^{2} \)
13 \( 1 - 2.51T + 2.19e3T^{2} \)
19 \( 1 + 14.3T + 6.85e3T^{2} \)
23 \( 1 - 180. iT - 1.21e4T^{2} \)
29 \( 1 + 41.2iT - 2.43e4T^{2} \)
31 \( 1 + 155. iT - 2.97e4T^{2} \)
37 \( 1 - 225. iT - 5.06e4T^{2} \)
41 \( 1 - 234. iT - 6.89e4T^{2} \)
43 \( 1 - 321.T + 7.95e4T^{2} \)
47 \( 1 + 326.T + 1.03e5T^{2} \)
53 \( 1 - 57.1T + 1.48e5T^{2} \)
59 \( 1 - 241.T + 2.05e5T^{2} \)
61 \( 1 - 460. iT - 2.26e5T^{2} \)
67 \( 1 - 392.T + 3.00e5T^{2} \)
71 \( 1 + 615. iT - 3.57e5T^{2} \)
73 \( 1 - 697. iT - 3.89e5T^{2} \)
79 \( 1 - 991. iT - 4.93e5T^{2} \)
83 \( 1 - 98.9T + 5.71e5T^{2} \)
89 \( 1 - 698.T + 7.04e5T^{2} \)
97 \( 1 - 1.42e3iT - 9.12e5T^{2} \)
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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

−11.72273268963202155585771738348, −11.25066970319747930555589414306, −9.749006153290916091263988699753, −9.132818821672584793104077311215, −8.376254137693051775754644973708, −6.07251853864174617257264668518, −5.41002713123303607422676392553, −4.75430004555868288816298471716, −3.69650496113929461394017775334, −2.71396453492388097574802738677, 0.59705203328241314511228675904, 2.24443599800574927184802794534, 3.45850953965141300477368623613, 4.73264597987928615512959507366, 6.18941234975000369629395926861, 6.87018287991276048690973899581, 7.43160884284183697853585198328, 8.703315409650282736112497439948, 10.45456779537334319974631782917, 11.24112987318200662644334851123

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