Properties

Label 2-11e2-11.3-c3-0-22
Degree $2$
Conductor $121$
Sign $-0.834 - 0.551i$
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  + (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

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.834 - 0.551i$
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: \((2,\ 121,\ (\ :3/2),\ -0.834 - 0.551i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.267360 + 0.888885i\)
\(L(\frac12)\) \(\approx\) \(0.267360 + 0.888885i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.08e3 - 791. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−12.30116957650838859477012161908, −11.84695894578520411992414334458, −11.01763950904457889998006129080, −8.766025772076802900021179495385, −7.918432335699735245122228367493, −6.84716652214070212210899814469, −5.28709608817165638603720893083, −4.07274174705893810407213793645, −2.28299947740226295908498580768, −0.39687397648272848317572567463, 3.72954752496905690904297636744, 4.20106109423146683416046633176, 5.56662477615206137443350964032, 6.64826663378829889024170167051, 8.069288637078392250859094068501, 9.756425622961613071310729114341, 10.50271752968581474754600475448, 11.35111419572107516753639408910, 12.54261906907941650946070936029, 14.05738938774043198239155209597

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