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Properties

Label 2-17e2-17.13-c1-0-8
Degree $2$
Conductor $289$
Sign $-0.185 + 0.982i$
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  − 1.53i·2-s + (−0.952 − 0.952i)3-s − 0.347·4-s + (2.49 + 2.49i)5-s + (−1.45 + 1.45i)6-s + (−0.245 + 0.245i)7-s − 2.53i·8-s − 1.18i·9-s + (3.82 − 3.82i)10-s + (1.24 − 1.24i)11-s + (0.330 + 0.330i)12-s + 3.29·13-s + (0.376 + 0.376i)14-s − 4.75i·15-s − 4.57·16-s + ⋯
L(s)  = 1  − 1.08i·2-s + (−0.550 − 0.550i)3-s − 0.173·4-s + (1.11 + 1.11i)5-s + (−0.595 + 0.595i)6-s + (−0.0928 + 0.0928i)7-s − 0.895i·8-s − 0.394i·9-s + (1.21 − 1.21i)10-s + (0.374 − 0.374i)11-s + (0.0955 + 0.0955i)12-s + 0.912·13-s + (0.100 + 0.100i)14-s − 1.22i·15-s − 1.14·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.185 + 0.982i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.185 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901907 - 1.08767i\)
\(L(\frac12)\) \(\approx\) \(0.901907 - 1.08767i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53iT - 2T^{2} \)
3 \( 1 + (0.952 + 0.952i)T + 3iT^{2} \)
5 \( 1 + (-2.49 - 2.49i)T + 5iT^{2} \)
7 \( 1 + (0.245 - 0.245i)T - 7iT^{2} \)
11 \( 1 + (-1.24 + 1.24i)T - 11iT^{2} \)
13 \( 1 - 3.29T + 13T^{2} \)
19 \( 1 + 1.53iT - 19T^{2} \)
23 \( 1 + (1.99 - 1.99i)T - 23iT^{2} \)
29 \( 1 + (0.837 + 0.837i)T + 29iT^{2} \)
31 \( 1 + (5.02 + 5.02i)T + 31iT^{2} \)
37 \( 1 + (-2.77 - 2.77i)T + 37iT^{2} \)
41 \( 1 + (-3.47 + 3.47i)T - 41iT^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 - 5.12T + 47T^{2} \)
53 \( 1 - 8.36iT - 53T^{2} \)
59 \( 1 + 10.8iT - 59T^{2} \)
61 \( 1 + (3.11 - 3.11i)T - 61iT^{2} \)
67 \( 1 - 8.07T + 67T^{2} \)
71 \( 1 + (7.95 + 7.95i)T + 71iT^{2} \)
73 \( 1 + (-1.20 - 1.20i)T + 73iT^{2} \)
79 \( 1 + (8.18 - 8.18i)T - 79iT^{2} \)
83 \( 1 - 2.14iT - 83T^{2} \)
89 \( 1 + 6.41T + 89T^{2} \)
97 \( 1 + (-2.88 - 2.88i)T + 97iT^{2} \)
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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

−11.14940521412685243662700698220, −11.10060978141742309121115447956, −9.845088517734198482488722437102, −9.206720782808800419832423744608, −7.37311874049621308603729197297, −6.31506381032109933254286045552, −5.97551070321057576824005820788, −3.76575763061040479788488850907, −2.64940853705651446079431576551, −1.36307335802021847247035339511, 1.88185309915278138711640719864, 4.30836301808095937464988420697, 5.41819879671696945720894619864, 5.83301591269519060623550966105, 6.99499035768776255166936517591, 8.307055631573751367725278562151, 9.049258718496963160967341044058, 10.11031475329229415069999901620, 10.98669473795471168930140694181, 12.11157470417046985582279548429

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