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Properties

Label 2-11e2-11.9-c1-0-2
Degree $2$
Conductor $121$
Sign $0.957 - 0.288i$
Analytic cond. $0.966189$
Root an. cond. $0.982949$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (−1.61 + 1.17i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (0.618 + 1.90i)6-s + (1.61 + 1.17i)7-s + (2.42 − 1.76i)8-s + (0.309 − 0.951i)9-s + 0.999·10-s − 2·12-s + (0.309 − 0.951i)13-s + (1.61 − 1.17i)14-s + (−1.61 − 1.17i)15-s + (−0.309 − 0.951i)16-s + (−1.54 − 4.75i)17-s + (−0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.934 + 0.678i)3-s + (0.404 + 0.293i)4-s + (0.138 + 0.425i)5-s + (0.252 + 0.776i)6-s + (0.611 + 0.444i)7-s + (0.858 − 0.623i)8-s + (0.103 − 0.317i)9-s + 0.316·10-s − 0.577·12-s + (0.0857 − 0.263i)13-s + (0.432 − 0.314i)14-s + (−0.417 − 0.303i)15-s + (−0.0772 − 0.237i)16-s + (−0.374 − 1.15i)17-s + (−0.190 − 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(0.966189\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06806 + 0.157149i\)
\(L(\frac12)\) \(\approx\) \(1.06806 + 0.157149i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.61 - 1.17i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.61 - 1.17i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.54 + 4.75i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.85 - 3.52i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (7.28 + 5.29i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.618 - 1.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.42 - 1.76i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.04 + 2.93i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (1.61 - 1.17i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.78 + 8.55i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.47 + 4.70i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.85 - 5.70i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-3.70 - 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.61 - 1.17i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.09 - 9.51i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.85 - 5.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (4.01 - 12.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.23682599145686504470344319694, −12.15306158010469197285956250938, −11.27254742197412040252295325203, −10.81663899455993554133169541934, −9.772481135402895666525453527886, −8.127549165379387071818685349803, −6.72358044996437021167768328671, −5.38829453357727510193616002343, −4.14399599651379544750664044509, −2.40901304053691265864152591971, 1.57641207120045443056074130854, 4.59285237723517242075396231430, 5.75168151247159767661265349320, 6.64957554437495276786418944110, 7.58088866062612657253939993670, 8.955451745361066875466482870905, 10.88912513036075784265298181053, 11.09712197155692508341047669278, 12.55017804083354804654940053392, 13.29953674693138346962129954771

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