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Properties

Label 2-242-11.4-c1-0-5
Degree $2$
Conductor $242$
Sign $0.0694 - 0.997i$
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 + (−0.618 + 1.90i)3-s + (0.309 + 0.951i)4-s + (2.42 − 1.76i)5-s + (−1.61 + 1.17i)6-s + (0.618 + 1.90i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 3·10-s − 2·12-s + (−4.04 − 2.93i)13-s + (−0.618 + 1.90i)14-s + (1.85 + 5.70i)15-s + (−0.809 + 0.587i)16-s + (−2.42 + 1.76i)17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.356 + 1.09i)3-s + (0.154 + 0.475i)4-s + (1.08 − 0.788i)5-s + (−0.660 + 0.479i)6-s + (0.233 + 0.718i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.948·10-s − 0.577·12-s + (−1.12 − 0.815i)13-s + (−0.165 + 0.508i)14-s + (0.478 + 1.47i)15-s + (−0.202 + 0.146i)16-s + (−0.588 + 0.427i)17-s + (−0.0728 − 0.224i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(1.93237\)
Root analytic conductor: \(1.39010\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :1/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26106 + 1.17628i\)
\(L(\frac12)\) \(\approx\) \(1.26106 + 1.17628i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 - 1.90i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.42 + 1.76i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.618 - 1.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.04 + 2.93i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.42 - 1.76i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.618 + 1.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (0.927 + 2.85i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.16 + 6.65i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.927 + 2.85i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-1.85 + 5.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.42 - 1.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.09 + 5.87i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + (9.70 - 7.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.32 - 13.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.61 + 1.17i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (14.5 - 10.5i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (8.89 + 6.46i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.62342848071758783409037612984, −11.38089176962191134957600775323, −10.35879031673709697455478371322, −9.415762555772107748681347616038, −8.719322281046440046108902404805, −7.21657132635131468010530192071, −5.55155722103691925451253315915, −5.36263886416648369404977620680, −4.25966695924330760611283280437, −2.43378586998150073411978031322, 1.53952071925937920679344294952, 2.73337637849958827908505694552, 4.56660635399976375239347471748, 5.87498837134768389062240052766, 6.88602651890828031752438519212, 7.31201145188749839980374073206, 9.250639473435220962422197464512, 10.20388577879237644315655996176, 11.08018320732199629905831387777, 11.98958141512235258879193291892

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