Properties

Label 2-11e2-11.4-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  + (−2.21 − 1.60i)2-s + (−2.44 + 7.54i)3-s + (−0.165 − 0.509i)4-s + (−12.0 + 8.73i)5-s + (17.5 − 12.7i)6-s + (0.949 + 2.92i)7-s + (−7.20 + 22.1i)8-s + (−29.0 − 21.0i)9-s + 40.5·10-s + 4.24·12-s + (−4.33 − 3.14i)13-s + (2.59 − 7.98i)14-s + (−36.3 − 112. i)15-s + (48.0 − 34.9i)16-s + (33.3 − 24.2i)17-s + (30.2 + 93.1i)18-s + ⋯
L(s)  = 1  + (−0.781 − 0.567i)2-s + (−0.471 + 1.45i)3-s + (−0.0207 − 0.0637i)4-s + (−1.07 + 0.781i)5-s + (1.19 − 0.866i)6-s + (0.0512 + 0.157i)7-s + (−0.318 + 0.980i)8-s + (−1.07 − 0.780i)9-s + 1.28·10-s + 0.102·12-s + (−0.0924 − 0.0672i)13-s + (0.0495 − 0.152i)14-s + (−0.626 − 1.92i)15-s + (0.751 − 0.545i)16-s + (0.475 − 0.345i)17-s + (0.396 + 1.21i)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: $\chi_{121} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :3/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0937835 - 0.117388i\)
\(L(\frac12)\) \(\approx\) \(0.0937835 - 0.117388i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.21 + 1.60i)T + (2.47 + 7.60i)T^{2} \)
3 \( 1 + (2.44 - 7.54i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (12.0 - 8.73i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-0.949 - 2.92i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (4.33 + 3.14i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-33.3 + 24.2i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-43.2 + 133. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 111.T + 1.21e4T^{2} \)
29 \( 1 + (7.72 + 23.7i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (25.4 + 18.5i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (-4.06 - 12.5i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-80.6 + 248. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 57.7T + 7.95e4T^{2} \)
47 \( 1 + (106. - 327. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-277. - 201. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-27.3 - 84.0i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (597. - 434. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 - 342.T + 3.00e5T^{2} \)
71 \( 1 + (-167. + 121. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (312. + 961. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (1.04e3 + 760. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (357. - 259. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + (1.08e3 + 791. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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

−11.92977024158741640413445202131, −11.30518671799113148713495114481, −10.60995290451484778428466732717, −9.760291948740097754811465328230, −8.833975963587008057750116806037, −7.44043964224867884320697950035, −5.62511967998897306588570240233, −4.38126842857172664856605856413, −2.97764375988463407796548786000, −0.12597261921877155317781230356, 1.20948828150126560026565782908, 3.88892692291587505519364114149, 5.82774946335877837845550251247, 7.07539549150320796984129851741, 7.932364025691760776064830730836, 8.351830902689068256864253355404, 9.919880657655484712997006859625, 11.64091412621198126555272000450, 12.31819793067250505686862148773, 12.86616635439403233722914635731

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