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Properties

Label 2-242-11.3-c1-0-4
Degree $2$
Conductor $242$
Sign $0.859 - 0.511i$
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  + (0.809 − 0.587i)2-s + (0.809 + 2.48i)3-s + (0.309 − 0.951i)4-s + (1 + 0.726i)5-s + (2.11 + 1.53i)6-s + (0.618 − 1.90i)7-s + (−0.309 − 0.951i)8-s + (−3.11 + 2.26i)9-s + 1.23·10-s + 2.61·12-s + (−2.61 + 1.90i)13-s + (−0.618 − 1.90i)14-s + (−1 + 3.07i)15-s + (−0.809 − 0.587i)16-s + (1.30 + 0.951i)17-s + (−1.19 + 3.66i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.467 + 1.43i)3-s + (0.154 − 0.475i)4-s + (0.447 + 0.324i)5-s + (0.864 + 0.628i)6-s + (0.233 − 0.718i)7-s + (−0.109 − 0.336i)8-s + (−1.03 + 0.755i)9-s + 0.390·10-s + 0.755·12-s + (−0.726 + 0.527i)13-s + (−0.165 − 0.508i)14-s + (−0.258 + 0.794i)15-s + (−0.202 − 0.146i)16-s + (0.317 + 0.230i)17-s + (−0.280 + 0.863i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90664 + 0.524394i\)
\(L(\frac12)\) \(\approx\) \(1.90664 + 0.524394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good3 \( 1 + (-0.809 - 2.48i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1 - 0.726i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.61 - 1.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.30 - 0.951i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.263 + 0.812i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.23T + 23T^{2} \)
29 \( 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3 + 9.23i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.04 + 3.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + (-0.763 - 2.35i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.47 + 6.15i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.97 - 6.06i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.23 - 3.80i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.0901T + 67T^{2} \)
71 \( 1 + (-0.618 - 0.449i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.89 - 12.0i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.8 - 7.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.11 + 3.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.09T + 89T^{2} \)
97 \( 1 + (11.2 - 8.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.05717476907753655681009291847, −11.04442193863887965202048835412, −10.13163324045146856990940772832, −9.812709393115662634898190517099, −8.572748680462971571180717010943, −7.13171615466677766962296537009, −5.70609321137377057384270336703, −4.51445603638552435177993498522, −3.80322869645990840255999087989, −2.39979923442256710516860608301, 1.80265991350573246380517491413, 3.02937557113702297031089403643, 5.02944613625552842122694455913, 5.98367482414487934934370539257, 7.03652027742650428547053725984, 7.946760194045910800547119645129, 8.702649818882557514345143099777, 9.967523282830812710681219337904, 11.68172474388461247062292754415, 12.26859009590965001025064901080

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