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Properties

Label 2-11e2-11.7-c2-0-3
Degree $2$
Conductor $121$
Sign $0.258 - 0.966i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.27 − 1.74i)2-s + (−0.881 + 2.71i)3-s + (−0.208 − 0.641i)4-s + (−6.18 + 4.49i)5-s + (3.62 + 4.98i)6-s + (−5.75 + 1.87i)7-s + (6.83 + 2.22i)8-s + (0.704 + 0.511i)9-s + 16.5i·10-s + 1.92·12-s + (−1.88 + 2.59i)13-s + (−4.04 + 12.4i)14-s + (−6.73 − 20.7i)15-s + (14.7 − 10.7i)16-s + (−3.66 − 5.03i)17-s + (1.79 − 0.581i)18-s + ⋯
L(s)  = 1  + (0.635 − 0.874i)2-s + (−0.293 + 0.903i)3-s + (−0.0521 − 0.160i)4-s + (−1.23 + 0.898i)5-s + (0.603 + 0.831i)6-s + (−0.822 + 0.267i)7-s + (0.854 + 0.277i)8-s + (0.0782 + 0.0568i)9-s + 1.65i·10-s + 0.160·12-s + (−0.145 + 0.199i)13-s + (−0.288 + 0.889i)14-s + (−0.448 − 1.38i)15-s + (0.922 − 0.670i)16-s + (−0.215 − 0.296i)17-s + (0.0994 − 0.0323i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.258 - 0.966i$
Analytic conductor: \(3.29701\)
Root analytic conductor: \(1.81576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1),\ 0.258 - 0.966i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03678 + 0.796242i\)
\(L(\frac12)\) \(\approx\) \(1.03678 + 0.796242i\)
\(L(2)\) 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 \( 1 + (-1.27 + 1.74i)T + (-1.23 - 3.80i)T^{2} \)
3 \( 1 + (0.881 - 2.71i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (6.18 - 4.49i)T + (7.72 - 23.7i)T^{2} \)
7 \( 1 + (5.75 - 1.87i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (1.88 - 2.59i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (3.66 + 5.03i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-18.4 - 6.01i)T + (292. + 212. i)T^{2} \)
23 \( 1 - 3.47T + 529T^{2} \)
29 \( 1 + (-34.3 + 11.1i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-2.66 - 1.93i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-19.5 - 60.1i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (30.3 + 9.86i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 43.9iT - 1.84e3T^{2} \)
47 \( 1 + (10.7 - 33.0i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (13.3 + 9.72i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (19.7 + 60.6i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-64.9 - 89.4i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 96.1T + 4.48e3T^{2} \)
71 \( 1 + (-36.2 + 26.3i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-60.1 + 19.5i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-8.29 + 11.4i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (52.0 + 71.6i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 51.3T + 7.92e3T^{2} \)
97 \( 1 + (25.4 + 18.5i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−13.25828385571273550470249479756, −12.05114695450662364624805618430, −11.50676240654751194038810062994, −10.55988872620534211654421523738, −9.750630561615539040017149066388, −7.967200174388123966756510381888, −6.81084923659289406211401085228, −4.90925435109319189016449850379, −3.80933204616294159051408497886, −2.95018436512933488875688244025, 0.827110782693363809839128858577, 3.85457096576645615591947429048, 5.09793907064835532613425117535, 6.47322317379436806522776545686, 7.27030160447071523342145004549, 8.178476337326366010870883360475, 9.754511681022455793983182282067, 11.29214916058700351418023355799, 12.45886546117385638727496047000, 12.90324788117654147372483280920

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