↑

Properties

Label 2-241-241.16-c1-0-5
Degree $2$
Conductor $241$
Sign $0.175 - 0.984i$
Analytic cond. $1.92439$
Root an. cond. $1.38722$
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  + (−0.415 + 0.719i)2-s + (0.269 + 0.466i)3-s + (0.654 + 1.13i)4-s + 1.25·5-s − 0.447·6-s + (3.75 + 2.16i)7-s − 2.75·8-s + (1.35 − 2.34i)9-s + (−0.520 + 0.901i)10-s + (−5.66 − 3.27i)11-s + (−0.352 + 0.610i)12-s + (1.62 − 0.936i)13-s + (−3.12 + 1.80i)14-s + (0.337 + 0.584i)15-s + (−0.165 + 0.286i)16-s + 6.72i·17-s + ⋯
L(s)  = 1  + (−0.293 + 0.509i)2-s + (0.155 + 0.269i)3-s + (0.327 + 0.566i)4-s + 0.560·5-s − 0.182·6-s + (1.41 + 0.819i)7-s − 0.972·8-s + (0.451 − 0.782i)9-s + (−0.164 + 0.285i)10-s + (−1.70 − 0.986i)11-s + (−0.101 + 0.176i)12-s + (0.449 − 0.259i)13-s + (−0.834 + 0.481i)14-s + (0.0871 + 0.150i)15-s + (−0.0413 + 0.0716i)16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(241\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(1.92439\)
Root analytic conductor: \(1.38722\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{241} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 241,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06287 + 0.889713i\)
\(L(\frac12)\) \(\approx\) \(1.06287 + 0.889713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad241 \( 1 + (11.5 - 10.3i)T \)
good2 \( 1 + (0.415 - 0.719i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.269 - 0.466i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.25T + 5T^{2} \)
7 \( 1 + (-3.75 - 2.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.66 + 3.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.62 + 0.936i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.72iT - 17T^{2} \)
19 \( 1 + (-2.50 - 1.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.44iT - 23T^{2} \)
29 \( 1 + (1.28 - 2.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.71 + 5.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.40 - 1.39i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.01T + 41T^{2} \)
43 \( 1 + 7.85iT - 43T^{2} \)
47 \( 1 - 2.97T + 47T^{2} \)
53 \( 1 + (3.77 + 6.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.720 + 1.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 2.14T + 61T^{2} \)
67 \( 1 + (-3.73 + 6.47i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.971 - 0.560i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.253iT - 73T^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 + (3.08 - 5.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-15.3 - 8.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.01 + 1.76i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.42429357512721013751411106166, −11.25234392656109558118625034443, −10.50764653140800489537000209775, −9.106012422484437820000295162280, −8.285492504562641742345479945720, −7.74937287559414867078878139053, −6.10020039240815083876812256340, −5.44540517942169863648851227784, −3.67244445096738497945333150751, −2.20481886195661558161452192392, 1.51173897599784979165083471930, 2.48367472075596855589742713550, 4.77093201917893523898380476318, 5.46771212604367504949154715375, 7.36812474039544913613699107416, 7.63222456382816001775214299994, 9.299268246554116637525997372056, 10.15449143323582101080470617363, 10.91549898815978641244560678653, 11.53874451701554785550092555762

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