Properties

Label 2-11e2-11.3-c3-0-9
Degree $2$
Conductor $121$
Sign $-0.220 - 0.975i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.592 − 0.430i)2-s + (1.83 + 5.63i)3-s + (−2.30 + 7.09i)4-s + (10.4 + 7.55i)5-s + (3.51 + 2.55i)6-s + (5.23 − 16.0i)7-s + (3.49 + 10.7i)8-s + (−6.58 + 4.78i)9-s + 9.41·10-s − 44.2·12-s + (−60.3 + 43.8i)13-s + (−3.82 − 11.7i)14-s + (−23.5 + 72.4i)15-s + (−41.6 − 30.2i)16-s + (66.9 + 48.6i)17-s + (−1.84 + 5.66i)18-s + ⋯
L(s)  = 1  + (0.209 − 0.152i)2-s + (0.352 + 1.08i)3-s + (−0.288 + 0.887i)4-s + (0.930 + 0.675i)5-s + (0.238 + 0.173i)6-s + (0.282 − 0.869i)7-s + (0.154 + 0.475i)8-s + (−0.244 + 0.177i)9-s + 0.297·10-s − 1.06·12-s + (−1.28 + 0.936i)13-s + (−0.0731 − 0.224i)14-s + (−0.405 + 1.24i)15-s + (−0.650 − 0.472i)16-s + (0.955 + 0.694i)17-s + (−0.0241 + 0.0742i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(7.13923\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :3/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.32987 + 1.66460i\)
\(L(\frac12)\) \(\approx\) \(1.32987 + 1.66460i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.592 + 0.430i)T + (2.47 - 7.60i)T^{2} \)
3 \( 1 + (-1.83 - 5.63i)T + (-21.8 + 15.8i)T^{2} \)
5 \( 1 + (-10.4 - 7.55i)T + (38.6 + 118. i)T^{2} \)
7 \( 1 + (-5.23 + 16.0i)T + (-277. - 201. i)T^{2} \)
13 \( 1 + (60.3 - 43.8i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-66.9 - 48.6i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (20.9 + 64.5i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 13.3T + 1.21e4T^{2} \)
29 \( 1 + (-52.2 + 160. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (-52.9 + 38.4i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-12.6 + 38.8i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (-84.9 - 261. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 2.28T + 7.95e4T^{2} \)
47 \( 1 + (-22.2 - 68.3i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (-120. + 87.5i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-168. + 518. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (81.9 + 59.5i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 411.T + 3.00e5T^{2} \)
71 \( 1 + (-380. - 276. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (-188. + 580. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (-791. + 574. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (21.1 + 15.3i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + 352.T + 7.04e5T^{2} \)
97 \( 1 + (685. - 498. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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

−13.51780838384768065402071178380, −12.28271681331038343736894330203, −11.02589219743709818179509366459, −10.00022014592336457447120926603, −9.382237648239006936012511313487, −7.929088101588973237555439035932, −6.72874087294531829015610068451, −4.83360414669785642417351681811, −3.91440242997707404642592776207, −2.54100542514602026683780887478, 1.12867717999534965530719790410, 2.36614030496348924347731834874, 5.10524557464090069440185987082, 5.68639523871673855980070030525, 7.11320882630493183166468093526, 8.372095822359339257371595801428, 9.487516158784775366853733817696, 10.33100421279199554646190377592, 12.21438983944428892680529790234, 12.71161818321182146576115517292

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