↑

Properties

Label 2-242-11.4-c1-0-6
Degree $2$
Conductor $242$
Sign $0.984 + 0.174i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.226 − 0.696i)3-s + (0.309 + 0.951i)4-s + (1.40 − 1.01i)5-s + (0.592 − 0.430i)6-s + (−1.46 − 4.50i)7-s + (−0.309 + 0.951i)8-s + (1.99 + 1.44i)9-s + 1.73·10-s + 0.732·12-s + (2.42 + 1.76i)13-s + (1.46 − 4.50i)14-s + (−0.391 − 1.20i)15-s + (−0.809 + 0.587i)16-s + (−4.20 + 3.05i)17-s + (0.761 + 2.34i)18-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.130 − 0.401i)3-s + (0.154 + 0.475i)4-s + (0.626 − 0.455i)5-s + (0.241 − 0.175i)6-s + (−0.552 − 1.70i)7-s + (−0.109 + 0.336i)8-s + (0.664 + 0.482i)9-s + 0.547·10-s + 0.211·12-s + (0.673 + 0.489i)13-s + (0.390 − 1.20i)14-s + (−0.101 − 0.311i)15-s + (−0.202 + 0.146i)16-s + (−1.01 + 0.740i)17-s + (0.179 + 0.552i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.984 + 0.174i$
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: \((2,\ 242,\ (\ :1/2),\ 0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83422 - 0.160870i\)
\(L(\frac12)\) \(\approx\) \(1.83422 - 0.160870i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.226 + 0.696i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.40 + 1.01i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.46 + 4.50i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.42 - 1.76i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.20 - 3.05i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.391 - 1.20i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.26T + 23T^{2} \)
29 \( 1 + (-0.927 - 2.85i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.158 + 0.115i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.22 + 6.84i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.248 + 0.764i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (2.53 - 7.79i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.05 + 6.58i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.92 - 9.00i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.05 + 1.49i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (-7.65 + 5.56i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.286 - 0.882i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.82 + 2.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.63 + 4.81i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.60794537613410794351000258870, −11.03948980726866497166387116419, −10.29317630225128128553052320979, −9.116244516633263995241386398120, −7.85941308317063456823199298254, −6.96329815533636730415082370216, −6.17695255945097260442329946833, −4.63561708355223024113744961855, −3.74547165245531295976842594433, −1.65607369973513518200613282070, 2.26361851230447954811862766272, 3.29333251285928587906823679272, 4.81109967634717334870840381799, 6.00171069143520838998706912123, 6.65793645982587837980295687845, 8.577087366393795703023286299704, 9.466088152554329811729744132752, 10.14203580351446851319192672156, 11.30577317169835265998749263806, 12.19861033425077945962915579304

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