Properties

Label 2-17e2-17.16-c1-0-0
Degree $2$
Conductor $289$
Sign $-0.911 - 0.410i$
Analytic cond. $2.30767$
Root an. cond. $1.51910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.53·2-s + 1.34i·3-s + 0.347·4-s + 3.53i·5-s − 2.06i·6-s + 0.347i·7-s + 2.53·8-s + 1.18·9-s − 5.41i·10-s + 1.75i·11-s + 0.467i·12-s − 3.29·13-s − 0.532i·14-s − 4.75·15-s − 4.57·16-s + ⋯
L(s)  = 1  − 1.08·2-s + 0.777i·3-s + 0.173·4-s + 1.57i·5-s − 0.842i·6-s + 0.131i·7-s + 0.895·8-s + 0.394·9-s − 1.71i·10-s + 0.530i·11-s + 0.135i·12-s − 0.912·13-s − 0.142i·14-s − 1.22·15-s − 1.14·16-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.911 - 0.410i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.911 - 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120476 + 0.560358i\)
\(L(\frac12)\) \(\approx\) \(0.120476 + 0.560358i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53T + 2T^{2} \)
3 \( 1 - 1.34iT - 3T^{2} \)
5 \( 1 - 3.53iT - 5T^{2} \)
7 \( 1 - 0.347iT - 7T^{2} \)
11 \( 1 - 1.75iT - 11T^{2} \)
13 \( 1 + 3.29T + 13T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 + 2.81iT - 23T^{2} \)
29 \( 1 + 1.18iT - 29T^{2} \)
31 \( 1 - 7.10iT - 31T^{2} \)
37 \( 1 + 3.92iT - 37T^{2} \)
41 \( 1 + 4.92iT - 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 5.12T + 47T^{2} \)
53 \( 1 - 8.36T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 - 4.41iT - 61T^{2} \)
67 \( 1 - 8.07T + 67T^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 - 1.70iT - 73T^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 - 2.14T + 83T^{2} \)
89 \( 1 - 6.41T + 89T^{2} \)
97 \( 1 - 4.08iT - 97T^{2} \)
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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

−11.84255026860971441132558350344, −10.66005984176071395569112770724, −10.28990565882236652392885284002, −9.662253700239910545347452963208, −8.584883553984798683206674406707, −7.31399589565976057666917371243, −6.82406349876047458226338411314, −5.02534235640028237099066424120, −3.82514743375886389301306111165, −2.28102035559970379373319182444, 0.63251848739655294396149255692, 1.80642340625490447407844020161, 4.27344633400243721553007143700, 5.28284287651621414997897738986, 6.83656809628690498277025332840, 7.904655181951897699703205250418, 8.408675709118140865423695966428, 9.445568274563134837174397657717, 10.04606212210840091644096849879, 11.46276440062832229473703189768

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