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Properties

Label 2-17e2-17.16-c3-0-12
Degree $2$
Conductor $289$
Sign $-0.242 - 0.970i$
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  − 1.36·2-s + 3.15i·3-s − 6.14·4-s + 3.03i·5-s − 4.29i·6-s + 7.94i·7-s + 19.2·8-s + 17.0·9-s − 4.12i·10-s − 27.6i·11-s − 19.3i·12-s + 58.1·13-s − 10.8i·14-s − 9.56·15-s + 22.9·16-s + ⋯
L(s)  = 1  − 0.481·2-s + 0.607i·3-s − 0.768·4-s + 0.271i·5-s − 0.292i·6-s + 0.428i·7-s + 0.851·8-s + 0.631·9-s − 0.130i·10-s − 0.756i·11-s − 0.466i·12-s + 1.23·13-s − 0.206i·14-s − 0.164·15-s + 0.358·16-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(1.102618692\)
\(L(\frac12)\) \(\approx\) \(1.102618692\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.36T + 8T^{2} \)
3 \( 1 - 3.15iT - 27T^{2} \)
5 \( 1 - 3.03iT - 125T^{2} \)
7 \( 1 - 7.94iT - 343T^{2} \)
11 \( 1 + 27.6iT - 1.33e3T^{2} \)
13 \( 1 - 58.1T + 2.19e3T^{2} \)
19 \( 1 + 89.1T + 6.85e3T^{2} \)
23 \( 1 - 115. iT - 1.21e4T^{2} \)
29 \( 1 + 128. iT - 2.43e4T^{2} \)
31 \( 1 - 273. iT - 2.97e4T^{2} \)
37 \( 1 + 132. iT - 5.06e4T^{2} \)
41 \( 1 - 470. iT - 6.89e4T^{2} \)
43 \( 1 + 352.T + 7.95e4T^{2} \)
47 \( 1 - 152.T + 1.03e5T^{2} \)
53 \( 1 + 527.T + 1.48e5T^{2} \)
59 \( 1 - 292.T + 2.05e5T^{2} \)
61 \( 1 - 53.8iT - 2.26e5T^{2} \)
67 \( 1 - 52.9T + 3.00e5T^{2} \)
71 \( 1 - 788. iT - 3.57e5T^{2} \)
73 \( 1 - 295. iT - 3.89e5T^{2} \)
79 \( 1 - 720. iT - 4.93e5T^{2} \)
83 \( 1 - 116.T + 5.71e5T^{2} \)
89 \( 1 + 813.T + 7.04e5T^{2} \)
97 \( 1 - 794. iT - 9.12e5T^{2} \)
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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

−11.26424762384787622747320412602, −10.58420407938391076149138435194, −9.715905013520099406092666372047, −8.804311719995356998010584704052, −8.197247949744697523660883719485, −6.76011794253626980558335324754, −5.52711115006523428537025609920, −4.36910547230199236139134875736, −3.36910359213516973796024266473, −1.28301001619530647601962916680, 0.59554342567872963154468413747, 1.78306775030053857824998439546, 3.93756554495922188285357588293, 4.77588508197250931115902352098, 6.35512178457433679296544000250, 7.32212617468382909501215614503, 8.311013773466760978929552572316, 9.038841601980414018665194395087, 10.18201732698352994419558493192, 10.80019608129341041583314830993

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