↑

Properties

Label 2-17e2-17.16-c3-0-44
Degree $2$
Conductor $289$
Sign $-0.970 - 0.242i$
Analytic cond. $17.0515$
Root an. cond. $4.12935$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.04·2-s − 2.60i·3-s + 17.4·4-s − 13.4i·5-s + 13.1i·6-s + 22.2i·7-s − 47.4·8-s + 20.1·9-s + 67.6i·10-s − 33.6i·11-s − 45.4i·12-s − 73.4·13-s − 111. i·14-s − 35.0·15-s + 100.·16-s + ⋯
L(s)  = 1  − 1.78·2-s − 0.502i·3-s + 2.17·4-s − 1.20i·5-s + 0.895i·6-s + 1.19i·7-s − 2.09·8-s + 0.747·9-s + 2.13i·10-s − 0.922i·11-s − 1.09i·12-s − 1.56·13-s − 2.13i·14-s − 0.602·15-s + 1.56·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.970 - 0.242i$
Analytic conductor: \(17.0515\)
Root analytic conductor: \(4.12935\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :3/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2634556292\)
\(L(\frac12)\) \(\approx\) \(0.2634556292\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.04T + 8T^{2} \)
3 \( 1 + 2.60iT - 27T^{2} \)
5 \( 1 + 13.4iT - 125T^{2} \)
7 \( 1 - 22.2iT - 343T^{2} \)
11 \( 1 + 33.6iT - 1.33e3T^{2} \)
13 \( 1 + 73.4T + 2.19e3T^{2} \)
19 \( 1 - 42.5T + 6.85e3T^{2} \)
23 \( 1 + 59.2iT - 1.21e4T^{2} \)
29 \( 1 - 21.3iT - 2.43e4T^{2} \)
31 \( 1 + 42.2iT - 2.97e4T^{2} \)
37 \( 1 + 265. iT - 5.06e4T^{2} \)
41 \( 1 - 80.0iT - 6.89e4T^{2} \)
43 \( 1 + 353.T + 7.95e4T^{2} \)
47 \( 1 - 52.4T + 1.03e5T^{2} \)
53 \( 1 + 551.T + 1.48e5T^{2} \)
59 \( 1 + 508.T + 2.05e5T^{2} \)
61 \( 1 - 671. iT - 2.26e5T^{2} \)
67 \( 1 + 859.T + 3.00e5T^{2} \)
71 \( 1 + 147. iT - 3.57e5T^{2} \)
73 \( 1 + 522. iT - 3.89e5T^{2} \)
79 \( 1 + 245. iT - 4.93e5T^{2} \)
83 \( 1 - 293.T + 5.71e5T^{2} \)
89 \( 1 + 72.0T + 7.04e5T^{2} \)
97 \( 1 - 1.38e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.58883378073023002454749952899, −9.506305800928353303372870684812, −9.033069290574245006846262901893, −8.148584240299249132058259552865, −7.39514794915099743538908174312, −6.18349571813790401128563941882, −4.95012504601617727351514276912, −2.56719858361110767948959387913, −1.41620354061914891077477217974, −0.18301736546250647593749867043, 1.62729445856647679490918227662, 3.07608566709929112600646700158, 4.64688531951277612130682933365, 6.78897954437709134282179284713, 7.18282112102800051259853147945, 7.84332381862671788018986000270, 9.542406613777555183210130902133, 9.940090457561257145069457295641, 10.46989265285318122259010946163, 11.29170748411136443347638760172

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