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Properties

Label 2-17e2-17.11-c2-0-19
Degree $2$
Conductor $289$
Sign $-0.563 - 0.825i$
Analytic cond. $7.87467$
Root an. cond. $2.80618$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.63 + 1.08i)2-s + (1.01 + 5.09i)3-s + (2.90 + 2.90i)4-s + (3.03 + 2.02i)5-s + (−2.88 + 14.5i)6-s + (2.74 − 1.83i)7-s + (0.119 + 0.288i)8-s + (−16.6 + 6.88i)9-s + (5.76 + 8.63i)10-s + (2.38 + 0.474i)11-s + (−11.8 + 17.7i)12-s + (7.73 − 7.73i)13-s + (9.22 − 1.83i)14-s + (−7.24 + 17.5i)15-s − 15.5i·16-s + ⋯
L(s)  = 1  + (1.31 + 0.544i)2-s + (0.337 + 1.69i)3-s + (0.726 + 0.726i)4-s + (0.606 + 0.405i)5-s + (−0.481 + 2.41i)6-s + (0.392 − 0.262i)7-s + (0.0149 + 0.0360i)8-s + (−1.84 + 0.765i)9-s + (0.576 + 0.863i)10-s + (0.216 + 0.0430i)11-s + (−0.988 + 1.47i)12-s + (0.594 − 0.594i)13-s + (0.658 − 0.131i)14-s + (−0.483 + 1.16i)15-s − 0.971i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.563 - 0.825i$
Analytic conductor: \(7.87467\)
Root analytic conductor: \(2.80618\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1),\ -0.563 - 0.825i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.78838 + 3.38692i\)
\(L(\frac12)\) \(\approx\) \(1.78838 + 3.38692i\)
\(L(2)\) 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 \( 1 + (-2.63 - 1.08i)T + (2.82 + 2.82i)T^{2} \)
3 \( 1 + (-1.01 - 5.09i)T + (-8.31 + 3.44i)T^{2} \)
5 \( 1 + (-3.03 - 2.02i)T + (9.56 + 23.0i)T^{2} \)
7 \( 1 + (-2.74 + 1.83i)T + (18.7 - 45.2i)T^{2} \)
11 \( 1 + (-2.38 - 0.474i)T + (111. + 46.3i)T^{2} \)
13 \( 1 + (-7.73 + 7.73i)T - 169iT^{2} \)
19 \( 1 + (8.45 + 3.50i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-0.674 + 3.38i)T + (-488. - 202. i)T^{2} \)
29 \( 1 + (-13.3 + 19.9i)T + (-321. - 776. i)T^{2} \)
31 \( 1 + (27.1 - 5.40i)T + (887. - 367. i)T^{2} \)
37 \( 1 + (-7.51 - 37.7i)T + (-1.26e3 + 523. i)T^{2} \)
41 \( 1 + (63.9 - 42.7i)T + (643. - 1.55e3i)T^{2} \)
43 \( 1 + (-50.0 + 20.7i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (-5.13 + 5.13i)T - 2.20e3iT^{2} \)
53 \( 1 + (-52.6 - 21.8i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-8.82 - 21.2i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (23.0 + 34.4i)T + (-1.42e3 + 3.43e3i)T^{2} \)
67 \( 1 + 17.3iT - 4.48e3T^{2} \)
71 \( 1 + (-10.0 - 50.6i)T + (-4.65e3 + 1.92e3i)T^{2} \)
73 \( 1 + (51.6 + 34.5i)T + (2.03e3 + 4.92e3i)T^{2} \)
79 \( 1 + (-116. - 23.1i)T + (5.76e3 + 2.38e3i)T^{2} \)
83 \( 1 + (-36.0 + 86.9i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (19.6 + 19.6i)T + 7.92e3iT^{2} \)
97 \( 1 + (8.46 - 12.6i)T + (-3.60e3 - 8.69e3i)T^{2} \)
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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

−11.92766031047413879671855542741, −10.81251557209979035496936719417, −10.15762473997350879551018760635, −9.226126975841939217661332176621, −8.076610168863014351743060731011, −6.52638461836583348168653664630, −5.60933689601308010359405803155, −4.68307154744164321251326632106, −3.87579263619161356651391054693, −2.83710615429671890596241761553, 1.51895179038319891500166004539, 2.28439179869593898245141234252, 3.72529026002536789852507759205, 5.28891635774020515907045234499, 6.08371234066800526434160052440, 7.07559905918798377479814238156, 8.362528379616952018652138068513, 9.080433565434462509486985728155, 10.88620638226753531172086936000, 11.75645576435185100338930279834

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