Properties

Label 2-546-91.59-c1-0-15
Degree $2$
Conductor $546$
Sign $0.0732 + 0.997i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + (0.707 + 0.707i)2-s + (−0.866 + 0.5i)3-s + 1.00i·4-s + (−1.42 − 0.382i)5-s + (−0.965 − 0.258i)6-s + (0.349 − 2.62i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.739 − 1.28i)10-s + (−5.63 − 1.51i)11-s + (−0.500 − 0.866i)12-s + (−2.68 − 2.41i)13-s + (2.10 − 1.60i)14-s + (1.42 − 0.382i)15-s − 1.00·16-s + 6.94·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.499 + 0.288i)3-s + 0.500i·4-s + (−0.638 − 0.171i)5-s + (−0.394 − 0.105i)6-s + (0.132 − 0.991i)7-s + (−0.250 + 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.233 − 0.404i)10-s + (−1.69 − 0.455i)11-s + (−0.144 − 0.250i)12-s + (−0.743 − 0.668i)13-s + (0.561 − 0.429i)14-s + (0.368 − 0.0987i)15-s − 0.250·16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0732 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0732 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0732 + 0.997i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0732 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506337 - 0.470503i\)
\(L(\frac12)\) \(\approx\) \(0.506337 - 0.470503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.349 + 2.62i)T \)
13 \( 1 + (2.68 + 2.41i)T \)
good5 \( 1 + (1.42 + 0.382i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (5.63 + 1.51i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 6.94T + 17T^{2} \)
19 \( 1 + (1.37 + 5.14i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.769iT - 23T^{2} \)
29 \( 1 + (4.35 - 7.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.101 + 0.379i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.14 + 4.14i)T - 37iT^{2} \)
41 \( 1 + (1.70 + 6.34i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.142 + 0.0825i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.23 + 12.0i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.44 - 7.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.616 - 0.616i)T + 59iT^{2} \)
61 \( 1 + (-4.89 - 2.82i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.23 - 4.61i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.36 - 5.08i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.94 - 2.39i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.96 + 6.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.29 - 9.29i)T - 83iT^{2} \)
89 \( 1 + (-6.30 - 6.30i)T + 89iT^{2} \)
97 \( 1 + (-0.852 - 0.228i)T + (84.0 + 48.5i)T^{2} \)
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

−10.61667032719612308804717391794, −9.995573386891773539033731680231, −8.542397851800367389900623835244, −7.54329052500876665601616025774, −7.27217759478324430867146266115, −5.61113778608919402343563673922, −5.13543628997235737582653200757, −4.02013317965774042954685030322, −2.96782362099514161978538598364, −0.34442123338273267153958013544, 1.95578321239629902702917705700, 3.06521475095947448569175146778, 4.48520069330489938764571763709, 5.40693057608805003867521193133, 6.12218458892114038725841011913, 7.66666846218815139649055310116, 7.967431801079099118849184873461, 9.647990425245496529950953363435, 10.17347757854929565866066633109, 11.32055778223548629949246021114

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