↑

Properties

Label 2-242-11.9-c3-0-18
Degree $2$
Conductor $242$
Sign $0.569 + 0.821i$
Analytic cond. $14.2784$
Root an. cond. $3.77868$
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  + (0.618 − 1.90i)2-s + (6.31 − 4.59i)3-s + (−3.23 − 2.35i)4-s + (4.60 + 14.1i)5-s + (−4.82 − 14.8i)6-s + (17.6 + 12.7i)7-s + (−6.47 + 4.70i)8-s + (10.5 − 32.3i)9-s + 29.8·10-s − 31.2·12-s + (13.6 − 41.8i)13-s + (35.2 − 25.5i)14-s + (94.2 + 68.4i)15-s + (4.94 + 15.2i)16-s + (7.69 + 23.6i)17-s + (−55.0 − 39.9i)18-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (1.21 − 0.883i)3-s + (−0.404 − 0.293i)4-s + (0.412 + 1.26i)5-s + (−0.328 − 1.01i)6-s + (0.950 + 0.690i)7-s + (−0.286 + 0.207i)8-s + (0.389 − 1.19i)9-s + 0.943·10-s − 0.751·12-s + (0.290 − 0.893i)13-s + (0.672 − 0.488i)14-s + (1.62 + 1.17i)15-s + (0.0772 + 0.237i)16-s + (0.109 + 0.338i)17-s + (−0.720 − 0.523i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(14.2784\)
Root analytic conductor: \(3.77868\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :3/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.01092 - 1.57607i\)
\(L(\frac12)\) \(\approx\) \(3.01092 - 1.57607i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.618 + 1.90i)T \)
11 \( 1 \)
good3 \( 1 + (-6.31 + 4.59i)T + (8.34 - 25.6i)T^{2} \)
5 \( 1 + (-4.60 - 14.1i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (-17.6 - 12.7i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (-13.6 + 41.8i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (-7.69 - 23.6i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-17.7 + 12.9i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 - 177.T + 1.21e4T^{2} \)
29 \( 1 + (120. + 87.8i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-23.2 + 71.4i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (179. + 130. i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (204. - 148. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 130.T + 7.95e4T^{2} \)
47 \( 1 + (403. - 293. i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (-3.99 + 12.3i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (28.7 + 20.9i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (166. + 511. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 + 519.T + 3.00e5T^{2} \)
71 \( 1 + (-24.2 - 74.6i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-925. - 672. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (238. - 734. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (166. + 510. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 - 667.T + 7.04e5T^{2} \)
97 \( 1 + (55.5 - 170. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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

−11.44932437687040154037627755972, −10.74823960778444861793961513144, −9.566172480838112255893413316141, −8.525894713919354663230943025779, −7.72419040293665293684009182314, −6.59824783698741229782390358476, −5.27928866631465858956714266306, −3.34680690307376841989866473271, −2.59197730674599220581679386785, −1.57279878293086002983449829293, 1.49827896261399512337842589304, 3.47890319263544293980466858536, 4.62910996432490173678394117137, 5.14240675191141204653059616482, 7.00695817950215174902690074974, 8.165829509403564056346104281233, 8.868767679223529877207429152674, 9.415785593386323875813506792218, 10.62013270829323471152509698306, 11.93121402171158533557289573376

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