Properties

Label 2-17e2-17.16-c3-0-56
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  + 1.58·2-s − 4.98i·3-s − 5.49·4-s − 14.4i·5-s − 7.88i·6-s − 29.4i·7-s − 21.3·8-s + 2.16·9-s − 22.8i·10-s + 13.5i·11-s + 27.4i·12-s + 45.7·13-s − 46.5i·14-s − 72.0·15-s + 10.2·16-s + ⋯
L(s)  = 1  + 0.559·2-s − 0.959i·3-s − 0.687·4-s − 1.29i·5-s − 0.536i·6-s − 1.58i·7-s − 0.943·8-s + 0.0800·9-s − 0.722i·10-s + 0.370i·11-s + 0.659i·12-s + 0.976·13-s − 0.888i·14-s − 1.23·15-s + 0.159·16-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

\(L(2)\) \(\approx\) \(1.526518186\)
\(L(\frac12)\) \(\approx\) \(1.526518186\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.58T + 8T^{2} \)
3 \( 1 + 4.98iT - 27T^{2} \)
5 \( 1 + 14.4iT - 125T^{2} \)
7 \( 1 + 29.4iT - 343T^{2} \)
11 \( 1 - 13.5iT - 1.33e3T^{2} \)
13 \( 1 - 45.7T + 2.19e3T^{2} \)
19 \( 1 + 113.T + 6.85e3T^{2} \)
23 \( 1 - 144. iT - 1.21e4T^{2} \)
29 \( 1 - 1.26iT - 2.43e4T^{2} \)
31 \( 1 - 30.0iT - 2.97e4T^{2} \)
37 \( 1 + 398. iT - 5.06e4T^{2} \)
41 \( 1 - 184. iT - 6.89e4T^{2} \)
43 \( 1 + 135.T + 7.95e4T^{2} \)
47 \( 1 - 247.T + 1.03e5T^{2} \)
53 \( 1 - 635.T + 1.48e5T^{2} \)
59 \( 1 + 625.T + 2.05e5T^{2} \)
61 \( 1 + 166. iT - 2.26e5T^{2} \)
67 \( 1 - 159.T + 3.00e5T^{2} \)
71 \( 1 + 19.4iT - 3.57e5T^{2} \)
73 \( 1 + 336. iT - 3.89e5T^{2} \)
79 \( 1 - 1.07e3iT - 4.93e5T^{2} \)
83 \( 1 + 47.1T + 5.71e5T^{2} \)
89 \( 1 + 626.T + 7.04e5T^{2} \)
97 \( 1 + 692. iT - 9.12e5T^{2} \)
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

−10.99529400924574309243665532609, −9.832730237908134618400379933058, −8.800216746034006396931794756380, −7.931948694822967481211866689628, −6.92748018831700170709072418072, −5.73116266586303545439091195424, −4.45235228617566331343022959884, −3.88139137240546840031022016946, −1.48289000723684138876471632601, −0.51966383622354098495824306191, 2.59322514157911689304347938675, 3.58980028228867299473614893517, 4.64578859211818381377610801725, 5.82461576467666298398880649703, 6.53605736689364904847291644653, 8.468434223105362237465561275974, 8.957187497344319346439975446123, 10.13785290018154918275349178628, 10.81660454747673837930706965505, 11.87136118311497550232562688118

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