↑

Properties

Label 2-209-19.7-c3-0-45
Degree $2$
Conductor $209$
Sign $-0.992 - 0.121i$
Analytic cond. $12.3313$
Root an. cond. $3.51160$
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  + (0.754 + 1.30i)2-s + (−2.95 − 5.11i)3-s + (2.86 − 4.95i)4-s + (−5.76 − 9.99i)5-s + (4.45 − 7.71i)6-s − 21.3·7-s + 20.7·8-s + (−3.91 + 6.78i)9-s + (8.70 − 15.0i)10-s + 11·11-s − 33.7·12-s + (1.93 − 3.34i)13-s + (−16.0 − 27.8i)14-s + (−34.0 + 58.9i)15-s + (−7.25 − 12.5i)16-s + (−1.46 − 2.53i)17-s + ⋯
L(s)  = 1  + (0.266 + 0.462i)2-s + (−0.567 − 0.983i)3-s + (0.357 − 0.619i)4-s + (−0.515 − 0.893i)5-s + (0.303 − 0.524i)6-s − 1.15·7-s + 0.915·8-s + (−0.145 + 0.251i)9-s + (0.275 − 0.476i)10-s + 0.301·11-s − 0.812·12-s + (0.0411 − 0.0713i)13-s + (−0.307 − 0.532i)14-s + (−0.586 + 1.01i)15-s + (−0.113 − 0.196i)16-s + (−0.0208 − 0.0361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $-0.992 - 0.121i$
Analytic conductor: \(12.3313\)
Root analytic conductor: \(3.51160\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :3/2),\ -0.992 - 0.121i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0483514 + 0.794374i\)
\(L(\frac12)\) \(\approx\) \(0.0483514 + 0.794374i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - 11T \)
19 \( 1 + (34.7 - 75.1i)T \)
good2 \( 1 + (-0.754 - 1.30i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (2.95 + 5.11i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (5.76 + 9.99i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + 21.3T + 343T^{2} \)
13 \( 1 + (-1.93 + 3.34i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (1.46 + 2.53i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (94.8 - 164. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-37.4 + 64.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 178.T + 2.97e4T^{2} \)
37 \( 1 + 395.T + 5.06e4T^{2} \)
41 \( 1 + (229. + 397. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-146. - 253. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-282. + 490. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (200. - 346. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (21.9 + 37.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-200. + 347. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-233. + 404. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-57.0 - 98.7i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (169. + 293. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (361. + 625. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 188.T + 5.71e5T^{2} \)
89 \( 1 + (613. - 1.06e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-398. - 690. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−11.92652483724098289713970739784, −10.46606584139728078209701659486, −9.484710830829868054636944668556, −8.094923809658664219101722356302, −7.03143697026807427538281949216, −6.26270120976928716599484769692, −5.41939529884380657067197902283, −3.88713243796521846220804574355, −1.63444362386801574640045240195, −0.32553122240690720704192773302, 2.69718446904007832427027432903, 3.68096628993251776116420431711, 4.61615693470041465226524671010, 6.37982597841186177321955631150, 7.10955812675752451168756490105, 8.548965019661784645559626754359, 9.975970402906830919996799496465, 10.60165675143833253839374835174, 11.35295452588439783653513348973, 12.22362742151377940680860624532

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