Properties

Label 2-11e2-11.9-c3-0-14
Degree $2$
Conductor $121$
Sign $0.263 + 0.964i$
Analytic cond. $7.13923$
Root an. cond. $2.67193$
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  + (−0.226 + 0.696i)2-s + (−4.79 + 3.48i)3-s + (6.03 + 4.38i)4-s + (−3.97 − 12.2i)5-s + (−1.34 − 4.12i)6-s + (−13.6 − 9.95i)7-s + (−9.15 + 6.65i)8-s + (2.51 − 7.74i)9-s + 9.41·10-s − 44.2·12-s + (23.0 − 70.9i)13-s + (10.0 − 7.28i)14-s + (61.6 + 44.7i)15-s + (15.8 + 48.9i)16-s + (−25.5 − 78.7i)17-s + (4.82 + 3.50i)18-s + ⋯
L(s)  = 1  + (−0.0799 + 0.246i)2-s + (−0.922 + 0.670i)3-s + (0.754 + 0.548i)4-s + (−0.355 − 1.09i)5-s + (−0.0912 − 0.280i)6-s + (−0.739 − 0.537i)7-s + (−0.404 + 0.294i)8-s + (0.0932 − 0.286i)9-s + 0.297·10-s − 1.06·12-s + (0.492 − 1.51i)13-s + (0.191 − 0.139i)14-s + (1.06 + 0.771i)15-s + (0.248 + 0.764i)16-s + (−0.364 − 1.12i)17-s + (0.0631 + 0.0458i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(7.13923\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :3/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.563124 - 0.429951i\)
\(L(\frac12)\) \(\approx\) \(0.563124 - 0.429951i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.226 - 0.696i)T + (-6.47 - 4.70i)T^{2} \)
3 \( 1 + (4.79 - 3.48i)T + (8.34 - 25.6i)T^{2} \)
5 \( 1 + (3.97 + 12.2i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (13.6 + 9.95i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (-23.0 + 70.9i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (25.5 + 78.7i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-54.9 + 39.9i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 - 13.3T + 1.21e4T^{2} \)
29 \( 1 + (136. + 99.3i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (20.2 - 62.2i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (33.0 + 24.0i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (222. - 161. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 2.28T + 7.95e4T^{2} \)
47 \( 1 + (58.1 - 42.2i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (46.0 - 141. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (441. + 320. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-31.3 - 96.3i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 - 411.T + 3.00e5T^{2} \)
71 \( 1 + (145. + 447. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (493. + 358. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (302. - 930. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-8.08 - 24.8i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + 352.T + 7.04e5T^{2} \)
97 \( 1 + (-261. + 806. i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−12.64835562386402282279096000756, −11.64507895992756581641445881947, −10.85662498265275840450669332782, −9.708945614947093901915641678612, −8.367258175845034565244324794352, −7.26124891607668762523466376458, −5.88594266144103788394554571030, −4.80941548048462734144191655125, −3.28365430240730959046458589908, −0.41413139738033480955694756895, 1.74658442665555663915037583988, 3.42118731669303917393884140558, 5.82200617313300127203682278126, 6.52608652902308103358975789572, 7.19141991350041409376732414077, 9.155470441107090568258038759372, 10.43237265475413915695617722032, 11.31858035081209135926322767350, 11.84217775698184350841442467059, 12.86610121325427754466146687604

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