↑

Properties

Label 2-546-273.254-c1-0-14
Degree $2$
Conductor $546$
Sign $0.993 - 0.109i$
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 + (−1.71 − 0.248i)3-s + (0.866 + 0.499i)4-s + (0.203 + 0.758i)5-s + (−1.59 − 0.683i)6-s + (−0.313 − 2.62i)7-s + (0.707 + 0.707i)8-s + (2.87 + 0.851i)9-s + 0.785i·10-s + (0.607 + 0.607i)11-s + (−1.36 − 1.07i)12-s + (2.66 + 2.43i)13-s + (0.376 − 2.61i)14-s + (−0.160 − 1.35i)15-s + (0.500 + 0.866i)16-s + (3.01 − 5.21i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.989 − 0.143i)3-s + (0.433 + 0.249i)4-s + (0.0909 + 0.339i)5-s + (−0.649 − 0.279i)6-s + (−0.118 − 0.992i)7-s + (0.249 + 0.249i)8-s + (0.958 + 0.283i)9-s + 0.248i·10-s + (0.183 + 0.183i)11-s + (−0.392 − 0.309i)12-s + (0.738 + 0.674i)13-s + (0.100 − 0.699i)14-s + (−0.0413 − 0.348i)15-s + (0.125 + 0.216i)16-s + (0.730 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.993 - 0.109i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.993 - 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72843 + 0.0950684i\)
\(L(\frac12)\) \(\approx\) \(1.72843 + 0.0950684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.248i)T \)
7 \( 1 + (0.313 + 2.62i)T \)
13 \( 1 + (-2.66 - 2.43i)T \)
good5 \( 1 + (-0.203 - 0.758i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.607 - 0.607i)T + 11iT^{2} \)
17 \( 1 + (-3.01 + 5.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.00 + 1.00i)T + 19iT^{2} \)
23 \( 1 + (-3.59 - 6.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.77 - 4.49i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.05 + 3.92i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.78 + 1.01i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.830 - 3.09i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.17 - 1.83i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.80 + 1.28i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-9.26 + 5.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (13.7 - 3.68i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + (7.26 + 7.26i)T + 67iT^{2} \)
71 \( 1 + (-1.26 - 0.338i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.49 + 0.400i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.32 + 4.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.36 + 1.36i)T - 83iT^{2} \)
89 \( 1 + (-0.800 + 2.98i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.64 - 6.14i)T + (-84.0 - 48.5i)T^{2} \)
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

−10.93790284071198940873377312840, −10.28181872806033983446779919396, −9.202682780147657786654682037158, −7.65003887964301715018759414141, −6.94644872481304703380195874653, −6.34677134019929638426092180939, −5.15530909542098980367938998753, −4.36145739649060810123095531856, −3.16477814002950658441188143187, −1.23409186001158137787513258513, 1.27895575280299778601314448995, 3.00895748300534551576372200420, 4.28292663526233101542973536426, 5.29391352508366582686236192631, 5.98837858713510482836089881318, 6.67630524817965946338759744806, 8.198728256020407665778741520962, 9.037782977893640807226917529491, 10.43470699980902819103630599374, 10.66091572074011571669121971657

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