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Properties

Label 2-11e2-11.4-c1-0-1
Degree $2$
Conductor $121$
Sign $0.220 - 0.975i$
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  + (1.61 + 1.17i)2-s + (−0.309 + 0.951i)3-s + (0.618 + 1.90i)4-s + (−0.809 + 0.587i)5-s + (−1.61 + 1.17i)6-s + (−0.618 − 1.90i)7-s + (1.61 + 1.17i)9-s − 2·10-s − 2.00·12-s + (−3.23 − 2.35i)13-s + (1.23 − 3.80i)14-s + (−0.309 − 0.951i)15-s + (3.23 − 2.35i)16-s + (1.61 − 1.17i)17-s + (1.23 + 3.80i)18-s + ⋯
L(s)  = 1  + (1.14 + 0.831i)2-s + (−0.178 + 0.549i)3-s + (0.309 + 0.951i)4-s + (−0.361 + 0.262i)5-s + (−0.660 + 0.479i)6-s + (−0.233 − 0.718i)7-s + (0.539 + 0.391i)9-s − 0.632·10-s − 0.577·12-s + (−0.897 − 0.652i)13-s + (0.330 − 1.01i)14-s + (−0.0797 − 0.245i)15-s + (0.809 − 0.587i)16-s + (0.392 − 0.285i)17-s + (0.291 + 0.896i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(0.966189\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1/2),\ 0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27164 + 1.01593i\)
\(L(\frac12)\) \(\approx\) \(1.27164 + 1.01593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 1.17i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.309 - 0.951i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.809 - 0.587i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.618 + 1.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.23 + 2.35i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.61 + 1.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.66 + 4.11i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.927 - 2.85i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.47 - 7.60i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-2.47 + 7.60i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.85 - 3.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.54 - 4.75i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.70 - 7.05i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + (-2.42 + 1.76i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.23 - 3.80i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.09 - 5.87i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.85 + 3.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (-5.66 - 4.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.68013789359199514990121804623, −13.04691549461484663704043115785, −11.87253528354608994981377983412, −10.50942291355663913681808412104, −9.740543239102578761272025647524, −7.68825012170275521608255626985, −7.07695754548104915286269228403, −5.58078696105245909706459075932, −4.56590598466553669105164376893, −3.48798401536756052280755262332, 2.07463231164244551369690537124, 3.72264831966698349248061952962, 4.97661972631937614006571407450, 6.26466009192094454467164952579, 7.65732721530845507138143271324, 9.139759794709922946000237174347, 10.45017620575187903567380704744, 11.84025812795032764660518085828, 12.22608483240759911383157992661, 12.88229033072322153899716239872

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