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Properties

Label 2-242-11.9-c1-0-5
Degree $2$
Conductor $242$
Sign $0.0783 + 0.996i$
Analytic cond. $1.93237$
Root an. cond. $1.39010$
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 + (2.21 − 1.60i)3-s + (−0.809 − 0.587i)4-s + (0.535 + 1.64i)5-s + (−0.844 − 2.59i)6-s + (−1.02 − 0.745i)7-s + (−0.809 + 0.587i)8-s + (1.37 − 4.24i)9-s + 1.73·10-s − 2.73·12-s + (0.927 − 2.85i)13-s + (−1.02 + 0.745i)14-s + (3.82 + 2.78i)15-s + (0.309 + 0.951i)16-s + (1.60 + 4.94i)17-s + (−3.61 − 2.62i)18-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (1.27 − 0.927i)3-s + (−0.404 − 0.293i)4-s + (0.239 + 0.736i)5-s + (−0.344 − 1.06i)6-s + (−0.387 − 0.281i)7-s + (−0.286 + 0.207i)8-s + (0.459 − 1.41i)9-s + 0.547·10-s − 0.788·12-s + (0.257 − 0.791i)13-s + (−0.274 + 0.199i)14-s + (0.988 + 0.718i)15-s + (0.0772 + 0.237i)16-s + (0.389 + 1.19i)17-s + (−0.851 − 0.618i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0783 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0783 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.0783 + 0.996i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :1/2),\ 0.0783 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39203 - 1.28688i\)
\(L(\frac12)\) \(\approx\) \(1.39203 - 1.28688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good3 \( 1 + (-2.21 + 1.60i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.535 - 1.64i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.02 + 0.745i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.927 + 2.85i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.60 - 4.94i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.82 - 2.78i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.73T + 23T^{2} \)
29 \( 1 + (-2.42 - 1.76i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.15 - 9.69i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.58 + 1.87i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.05 + 6.58i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (1.77 - 1.29i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.248 + 0.764i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.05 + 1.49i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.92 - 9.00i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.196T + 67T^{2} \)
71 \( 1 + (0.783 + 2.41i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.4 + 7.59i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.391 - 1.20i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.678 + 2.08i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 + (0.309 - 0.951i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.38607222993929264156319394293, −10.65632618734401596951228913383, −10.22488359574971294068079023713, −8.835776153971817470051661361797, −8.095766608318707661389420751755, −6.96404254295914970468905820442, −5.92484094783559857134616937902, −3.82440712981948407638679891244, −2.95024897174784898936876747323, −1.73104834410685240683643868214, 2.59416430774257530313069822620, 4.00375537560500480305433016009, 4.83963769962810852324892778544, 6.20028895473836000151698479044, 7.62706293261133120692539888151, 8.626755678212497750582367165674, 9.295343665868021158404219483714, 9.842442031728801088797830319960, 11.40546638630944326637382326579, 12.70612647298895795690038359598

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