↑

Properties

Label 2-546-39.32-c1-0-7
Degree $2$
Conductor $546$
Sign $0.383 - 0.923i$
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  + (0.965 − 0.258i)2-s + (0.442 + 1.67i)3-s + (0.866 − 0.499i)4-s + (0.359 + 0.359i)5-s + (0.860 + 1.50i)6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−2.60 + 1.48i)9-s + (0.440 + 0.254i)10-s + (0.529 + 1.97i)11-s + (1.22 + 1.22i)12-s + (−0.322 + 3.59i)13-s + i·14-s + (−0.442 + 0.760i)15-s + (0.500 − 0.866i)16-s + (0.334 + 0.579i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.255 + 0.966i)3-s + (0.433 − 0.249i)4-s + (0.160 + 0.160i)5-s + (0.351 + 0.613i)6-s + (−0.0978 + 0.365i)7-s + (0.249 − 0.249i)8-s + (−0.869 + 0.493i)9-s + (0.139 + 0.0803i)10-s + (0.159 + 0.595i)11-s + (0.352 + 0.354i)12-s + (−0.0895 + 0.995i)13-s + 0.267i·14-s + (−0.114 + 0.196i)15-s + (0.125 − 0.216i)16-s + (0.0810 + 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92201 + 1.28284i\)
\(L(\frac12)\) \(\approx\) \(1.92201 + 1.28284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (-0.442 - 1.67i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (0.322 - 3.59i)T \)
good5 \( 1 + (-0.359 - 0.359i)T + 5iT^{2} \)
11 \( 1 + (-0.529 - 1.97i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.334 - 0.579i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 0.834i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.54 + 6.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.59 - 1.49i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.15 - 4.15i)T - 31iT^{2} \)
37 \( 1 + (4.74 - 1.27i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.56 + 1.49i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.09 - 2.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.15 + 9.15i)T - 47iT^{2} \)
53 \( 1 - 2.70iT - 53T^{2} \)
59 \( 1 + (1.53 + 0.411i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.05 - 1.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.55 + 13.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.72 + 13.9i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.50 - 2.50i)T + 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + (-2.91 - 2.91i)T + 83iT^{2} \)
89 \( 1 + (3.28 + 12.2i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-16.9 - 4.54i)T + (84.0 + 48.5i)T^{2} \)
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

−10.85067018593972831048059539583, −10.23124262098497479209580190051, −9.302593546823461503986104215492, −8.573217577329871005425257246531, −7.18429646761563355255304591482, −6.22301004708052071176309859029, −5.09204965053302488046773492591, −4.37143386303886479747573177668, −3.25338175990747526692147299035, −2.14696339845218834267112168169, 1.15586130697502338349337117097, 2.79234131900443411612012574516, 3.67206290775084177364317749284, 5.32592439013478100220227012566, 5.90049764667389378748591069196, 7.15937039926772869186449425153, 7.61144671407181061744110343118, 8.695017857740535912342527019909, 9.655644123647164920407155701037, 10.98819524100079953228703217688

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