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Properties

Label 2-17e2-17.12-c2-0-9
Degree $2$
Conductor $289$
Sign $-0.110 + 0.993i$
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  + (−1.30 − 3.15i)2-s + (−3.14 + 2.10i)3-s + (−5.40 + 5.40i)4-s + (−3.98 − 0.793i)5-s + (10.7 + 7.17i)6-s + (−7.05 + 1.40i)7-s + (11.4 + 4.74i)8-s + (2.03 − 4.91i)9-s + (2.70 + 13.6i)10-s + (−5.66 + 8.47i)11-s + (5.63 − 28.3i)12-s + (6.16 + 6.16i)13-s + (13.6 + 20.4i)14-s + (14.2 − 5.88i)15-s − 11.7i·16-s + ⋯
L(s)  = 1  + (−0.652 − 1.57i)2-s + (−1.04 + 0.700i)3-s + (−1.35 + 1.35i)4-s + (−0.797 − 0.158i)5-s + (1.78 + 1.19i)6-s + (−1.00 + 0.200i)7-s + (1.43 + 0.593i)8-s + (0.226 − 0.546i)9-s + (0.270 + 1.36i)10-s + (−0.515 + 0.770i)11-s + (0.469 − 2.36i)12-s + (0.474 + 0.474i)13-s + (0.973 + 1.45i)14-s + (0.947 − 0.392i)15-s − 0.736i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.110 + 0.993i$
Analytic conductor: \(7.87467\)
Root analytic conductor: \(2.80618\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1),\ -0.110 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.180661 - 0.201825i\)
\(L(\frac12)\) \(\approx\) \(0.180661 - 0.201825i\)
\(L(2)\) 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 \( 1 + (1.30 + 3.15i)T + (-2.82 + 2.82i)T^{2} \)
3 \( 1 + (3.14 - 2.10i)T + (3.44 - 8.31i)T^{2} \)
5 \( 1 + (3.98 + 0.793i)T + (23.0 + 9.56i)T^{2} \)
7 \( 1 + (7.05 - 1.40i)T + (45.2 - 18.7i)T^{2} \)
11 \( 1 + (5.66 - 8.47i)T + (-46.3 - 111. i)T^{2} \)
13 \( 1 + (-6.16 - 6.16i)T + 169iT^{2} \)
19 \( 1 + (12.4 + 30.1i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (-1.02 - 0.682i)T + (202. + 488. i)T^{2} \)
29 \( 1 + (4.44 - 22.3i)T + (-776. - 321. i)T^{2} \)
31 \( 1 + (15.3 + 22.9i)T + (-367. + 887. i)T^{2} \)
37 \( 1 + (-35.4 + 23.6i)T + (523. - 1.26e3i)T^{2} \)
41 \( 1 + (-21.1 + 4.21i)T + (1.55e3 - 643. i)T^{2} \)
43 \( 1 + (9.49 - 22.9i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (-44.1 - 44.1i)T + 2.20e3iT^{2} \)
53 \( 1 + (-32.5 - 78.5i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (36.1 + 14.9i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (1.44 + 7.26i)T + (-3.43e3 + 1.42e3i)T^{2} \)
67 \( 1 + 99.8iT - 4.48e3T^{2} \)
71 \( 1 + (-3.91 + 2.61i)T + (1.92e3 - 4.65e3i)T^{2} \)
73 \( 1 + (-100. - 20.0i)T + (4.92e3 + 2.03e3i)T^{2} \)
79 \( 1 + (-32.6 + 48.9i)T + (-2.38e3 - 5.76e3i)T^{2} \)
83 \( 1 + (-83.6 + 34.6i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-5.23 + 5.23i)T - 7.92e3iT^{2} \)
97 \( 1 + (18.9 - 95.4i)T + (-8.69e3 - 3.60e3i)T^{2} \)
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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

−10.98967489375849757944146701694, −10.81884344873321789554300963048, −9.582496248697656095610851193807, −9.120485596454035339913051654467, −7.71920713340357442357602211635, −6.25862033202539034355231158656, −4.70556810374193130319015821873, −3.90521530001670061771951742394, −2.51822912406968605176548122036, −0.37962840564317968374458050848, 0.59539120217529803630606672898, 3.68727655623129090181705256669, 5.51344838668735262767272116176, 6.10242536851382715743623895555, 6.87595500667566073654550921963, 7.79808368915692237809554038585, 8.497300460838743850160057646709, 9.842427079300425686735175110054, 10.77549491192099763825804486971, 11.85637151343304015269998281443

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