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Properties

Label 2-242-11.5-c1-0-3
Degree $2$
Conductor $242$
Sign $-0.353 - 0.935i$
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 + (1.61 + 1.17i)3-s + (−0.809 + 0.587i)4-s + (−0.927 + 2.85i)5-s + (−0.618 + 1.90i)6-s + (1.61 − 1.17i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s − 3·10-s − 1.99·12-s + (−1.54 − 4.75i)13-s + (1.61 + 1.17i)14-s + (−4.85 + 3.52i)15-s + (0.309 − 0.951i)16-s + (−0.927 + 2.85i)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 + (−0.414 + 1.27i)5-s + (−0.252 + 0.776i)6-s + (0.611 − 0.444i)7-s + (−0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s − 0.948·10-s − 0.577·12-s + (−0.428 − 1.31i)13-s + (0.432 + 0.314i)14-s + (−1.25 + 0.910i)15-s + (0.0772 − 0.237i)16-s + (−0.224 + 0.691i)17-s + (−0.190 + 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $-0.353 - 0.935i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :1/2),\ -0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.955363 + 1.38292i\)
\(L(\frac12)\) \(\approx\) \(0.955363 + 1.38292i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 1.17i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.927 - 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.54 + 4.75i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.927 - 2.85i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.61 - 1.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (2.42 - 1.76i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.618 - 1.90i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.66 + 4.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.42 - 1.76i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (4.85 + 3.52i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.927 + 2.85i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.09 + 9.51i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + (-3.70 + 11.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-11.3 + 8.22i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.618 + 1.90i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.56 - 17.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-3.39 - 10.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.61746976380851089621741350085, −11.18367431417662814844170319665, −10.46369864450507313751878386760, −9.462122227285091575015017178890, −8.218442333592887859121097418127, −7.62174415354649277039938068051, −6.54217077184994747755515742215, −5.05307937331263625157316216831, −3.73131885455150224204099152280, −2.95087997929376551508478116347, 1.44000740456367459066609762560, 2.68597181236675632694368193775, 4.38789702618575015272993867864, 5.18916228497160214687033864297, 7.05893944788126403196303485209, 8.171323434084374599400630485883, 8.893999029868460559083859194618, 9.540104176372988698471333158807, 11.35056717122604040130312908538, 11.82025144157246570923784666544

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