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Properties

Label 2-242-11.5-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

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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: $\chi_{242} (27, \cdot )$
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 \( 1 + (55.5 + 170. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

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

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