Properties

Label 2-546-91.24-c1-0-4
Degree $2$
Conductor $546$
Sign $0.243 - 0.970i$
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 + i·3-s + (0.866 − 0.499i)4-s + (−1.45 − 0.391i)5-s + (0.258 + 0.965i)6-s + (0.541 + 2.58i)7-s + (0.707 − 0.707i)8-s − 9-s − 1.51·10-s + (−4.40 + 4.40i)11-s + (0.499 + 0.866i)12-s + (2.30 + 2.77i)13-s + (1.19 + 2.36i)14-s + (0.391 − 1.45i)15-s + (0.500 − 0.866i)16-s + (3.75 + 6.50i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (−0.652 − 0.174i)5-s + (0.105 + 0.394i)6-s + (0.204 + 0.978i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s − 0.477·10-s + (−1.32 + 1.32i)11-s + (0.144 + 0.249i)12-s + (0.638 + 0.769i)13-s + (0.318 + 0.631i)14-s + (0.101 − 0.376i)15-s + (0.125 − 0.216i)16-s + (0.910 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.243 - 0.970i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.243 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41973 + 1.10791i\)
\(L(\frac12)\) \(\approx\) \(1.41973 + 1.10791i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (-0.541 - 2.58i)T \)
13 \( 1 + (-2.30 - 2.77i)T \)
good5 \( 1 + (1.45 + 0.391i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.40 - 4.40i)T - 11iT^{2} \)
17 \( 1 + (-3.75 - 6.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.46 + 5.46i)T - 19iT^{2} \)
23 \( 1 + (0.0523 + 0.0302i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.60 - 2.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.10 + 7.86i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.25 - 4.66i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.84 + 1.03i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.84 - 2.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.335 + 1.25i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.81 - 3.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.57 + 9.61i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 0.829iT - 61T^{2} \)
67 \( 1 + (0.615 + 0.615i)T + 67iT^{2} \)
71 \( 1 + (0.497 - 0.133i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.87 + 2.10i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.73 + 9.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.74 + 4.74i)T - 83iT^{2} \)
89 \( 1 + (2.77 - 0.742i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.663 + 2.47i)T + (-84.0 + 48.5i)T^{2} \)
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

−11.17490319469372778697484738666, −10.18463825371876830752356194843, −9.380084062845401141766679486925, −8.254310219348722741705155714868, −7.48560205963578835228172021123, −6.11880757061548452003604377224, −5.17044089868205174871876521313, −4.44440181784423889345439043538, −3.30420840613077076635534161346, −2.05219262351380924335131425055, 0.871023657166359607932829216395, 3.03450924915542492477209992718, 3.60466145666392959318305516773, 5.21389856787849314009500700414, 5.79888678981876628584625277332, 7.24336625213693738902018094245, 7.70190405129918243525676344652, 8.348781777188672937240206796280, 10.00629971937551103801398169341, 10.86805781590262166590706825412

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