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Properties

Label 2-242-11.3-c1-0-6
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 + (0.118 + 0.363i)3-s + (0.309 − 0.951i)4-s + (−2.61 − 1.90i)5-s + (0.309 + 0.224i)6-s + (0.618 − 1.90i)7-s + (−0.309 − 0.951i)8-s + (2.30 − 1.67i)9-s − 3.23·10-s + 0.381·12-s + (1 − 0.726i)13-s + (−0.618 − 1.90i)14-s + (0.381 − 1.17i)15-s + (−0.809 − 0.587i)16-s + (−0.5 − 0.363i)17-s + (0.881 − 2.71i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.0681 + 0.209i)3-s + (0.154 − 0.475i)4-s + (−1.17 − 0.850i)5-s + (0.126 + 0.0916i)6-s + (0.233 − 0.718i)7-s + (−0.109 − 0.336i)8-s + (0.769 − 0.559i)9-s − 1.02·10-s + 0.110·12-s + (0.277 − 0.201i)13-s + (−0.165 − 0.508i)14-s + (0.0986 − 0.303i)15-s + (−0.202 − 0.146i)16-s + (−0.121 − 0.0881i)17-s + (0.207 − 0.639i)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: $\chi_{242} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :1/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08767 - 1.01455i\)
\(L(\frac12)\) \(\approx\) \(1.08767 - 1.01455i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.118 - 0.363i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.61 + 1.90i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1 + 0.726i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.5 + 0.363i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.80 - 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.23T + 23T^{2} \)
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 \( 1 + (1.14 - 3.52i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.73 + 5.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.56T + 43T^{2} \)
47 \( 1 + (2 + 6.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.23 + 0.898i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2 + 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + (-4.23 - 3.07i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.20 - 9.87i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.8 + 7.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.54 - 5.48i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 + (5.78 - 4.20i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.13391597230518203999324649006, −11.04792708763238496898337874890, −10.17165186982091714835247829450, −9.020455933836662506220168654293, −7.903428273838257440654242891503, −6.94301287702127475648664203303, −5.32924461674939101775743649597, −4.18850573822963334137827279339, −3.61651189708445922294139290111, −1.15064692771825852063622843627, 2.53945534883162019605907442548, 3.88969746326576793561985612984, 4.99285091912796351877255536375, 6.44675294999558231433764344432, 7.36609618030235790531557392809, 8.029387781688264784204851226824, 9.285461830873707087141197828003, 10.85628774414615155413395713435, 11.42657731741718537053885591501, 12.36729479989783727031224790626

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