↑

Properties

Label 2-11e2-11.6-c2-0-12
Degree $2$
Conductor $121$
Sign $-0.624 + 0.781i$
Analytic cond. $3.29701$
Root an. cond. $1.81576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.690 − 0.224i)2-s + (−1.11 − 0.812i)3-s + (−2.80 + 2.04i)4-s + (1.23 − 3.80i)5-s + (−0.954 − 0.310i)6-s + (−5.85 − 8.05i)7-s + (−3.19 + 4.39i)8-s + (−2.19 − 6.74i)9-s − 2.90i·10-s + 4.79·12-s + (5 − 1.62i)13-s + (−5.85 − 4.25i)14-s + (−4.47 + 3.24i)15-s + (3.07 − 9.45i)16-s + (−14.5 − 4.72i)17-s + (−3.02 − 4.16i)18-s + ⋯
L(s)  = 1  + (0.345 − 0.112i)2-s + (−0.372 − 0.270i)3-s + (−0.702 + 0.510i)4-s + (0.247 − 0.760i)5-s + (−0.159 − 0.0517i)6-s + (−0.836 − 1.15i)7-s + (−0.398 + 0.549i)8-s + (−0.243 − 0.749i)9-s − 0.290i·10-s + 0.399·12-s + (0.384 − 0.124i)13-s + (−0.418 − 0.303i)14-s + (−0.298 + 0.216i)15-s + (0.192 − 0.591i)16-s + (−0.854 − 0.277i)17-s + (−0.168 − 0.231i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.624 + 0.781i$
Analytic conductor: \(3.29701\)
Root analytic conductor: \(1.81576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1),\ -0.624 + 0.781i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.359333 - 0.747003i\)
\(L(\frac12)\) \(\approx\) \(0.359333 - 0.747003i\)
\(L(2)\) 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 \( 1 + (-0.690 + 0.224i)T + (3.23 - 2.35i)T^{2} \)
3 \( 1 + (1.11 + 0.812i)T + (2.78 + 8.55i)T^{2} \)
5 \( 1 + (-1.23 + 3.80i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (5.85 + 8.05i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (-5 + 1.62i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (14.5 + 4.72i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (1.21 - 1.67i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 2.76T + 529T^{2} \)
29 \( 1 + (-16.7 - 22.9i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (2.20 + 6.77i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-32.5 + 23.6i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (-41.2 + 56.7i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 23.0iT - 1.84e3T^{2} \)
47 \( 1 + (22.0 + 16.0i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (3.54 + 10.8i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-1.83 + 1.33i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (21.5 + 6.98i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 38.4T + 4.48e3T^{2} \)
71 \( 1 + (-23.5 + 72.5i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (60.4 + 83.2i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-3.74 + 1.21i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (79.1 + 25.7i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 123.T + 7.92e3T^{2} \)
97 \( 1 + (23.9 + 73.6i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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

−12.95062386028992778481593199958, −12.18239786071901136641745088748, −10.90413672817855507272288258828, −9.501517805512843883425640158096, −8.745582630406991311944118426021, −7.25976241451921343675334352716, −6.01310351686391826383338736540, −4.57908854372368418963488339146, −3.42946491533658633372279325096, −0.55138576299877891659376571212, 2.70561926883744549884225284273, 4.49998527093393268084731057071, 5.82231144234015825621235253035, 6.44955424899233678905969332013, 8.435642315235203213185137943395, 9.510517191368277961929243165616, 10.40988476114019193673345720717, 11.45376676020339793881144994186, 12.78464941864639168354369918417, 13.58186529701167648328630865163

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