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Properties

Label 2-17e2-17.15-c1-0-12
Degree $2$
Conductor $289$
Sign $0.897 + 0.440i$
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  + (0.707 + 0.707i)2-s − 0.999i·4-s + (1.84 − 0.765i)5-s + (−3.69 − 1.53i)7-s + (2.12 − 2.12i)8-s + (2.12 − 2.12i)9-s + (1.84 + 0.765i)10-s + 2i·13-s + (−1.53 − 3.69i)14-s + 1.00·16-s + 3·18-s + (2.82 + 2.82i)19-s + (−0.765 − 1.84i)20-s + (−1.53 + 3.69i)23-s + (−0.707 + 0.707i)25-s + (−1.41 + 1.41i)26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s − 0.499i·4-s + (0.826 − 0.342i)5-s + (−1.39 − 0.578i)7-s + (0.750 − 0.750i)8-s + (0.707 − 0.707i)9-s + (0.584 + 0.242i)10-s + 0.554i·13-s + (−0.409 − 0.987i)14-s + 0.250·16-s + 0.707·18-s + (0.648 + 0.648i)19-s + (−0.171 − 0.413i)20-s + (−0.319 + 0.770i)23-s + (−0.141 + 0.141i)25-s + (−0.277 + 0.277i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.897 + 0.440i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69508 - 0.393606i\)
\(L(\frac12)\) \(\approx\) \(1.69508 - 0.393606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T + 2iT^{2} \)
3 \( 1 + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.84 + 0.765i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.69 + 1.53i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
19 \( 1 + (-2.82 - 2.82i)T + 19iT^{2} \)
23 \( 1 + (1.53 - 3.69i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-5.54 + 2.29i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.53 - 3.69i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.765 - 1.84i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.54 - 2.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (2.82 - 2.82i)T - 43iT^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (4.24 + 4.24i)T + 53iT^{2} \)
59 \( 1 + (8.48 - 8.48i)T - 59iT^{2} \)
61 \( 1 + (9.23 + 3.82i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-1.53 - 3.69i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.54 + 2.29i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.59 + 11.0i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (2.82 + 2.82i)T + 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (-1.84 + 0.765i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.01888388080435116505550371147, −10.44675224850484486760877295703, −9.724404684870939526973597613610, −9.362639846166966251165329682919, −7.47700759672826145854373600929, −6.48037314631550749201181126306, −6.00904726822449727725868468646, −4.65676970058787066027116398630, −3.49930018542454127824634617179, −1.32131578152751341104153425809, 2.32441696413495850170768370136, 3.14326891294711734362242683262, 4.58272573813362167647593560266, 5.83075769406412745492111580351, 6.86359477942518334257422124334, 7.992121603139234272860681602011, 9.265667215801043140615677806613, 10.12099155008720173599196864988, 10.88836303992088618071021739990, 12.21124890294023688643285779846

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