Properties

Label 2-546-91.80-c1-0-4
Degree $2$
Conductor $546$
Sign $0.999 + 0.0139i$
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  + (−0.258 − 0.965i)2-s i·3-s + (−0.866 + 0.499i)4-s + (−0.349 + 1.30i)5-s + (−0.965 + 0.258i)6-s + (−0.635 + 2.56i)7-s + (0.707 + 0.707i)8-s − 9-s + 1.35·10-s + (−1.88 − 1.88i)11-s + (0.499 + 0.866i)12-s + (3.33 − 1.36i)13-s + (2.64 − 0.0509i)14-s + (1.30 + 0.349i)15-s + (0.500 − 0.866i)16-s + (2.62 + 4.54i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s − 0.577i·3-s + (−0.433 + 0.249i)4-s + (−0.156 + 0.583i)5-s + (−0.394 + 0.105i)6-s + (−0.240 + 0.970i)7-s + (0.249 + 0.249i)8-s − 0.333·9-s + 0.427·10-s + (−0.567 − 0.567i)11-s + (0.144 + 0.249i)12-s + (0.925 − 0.379i)13-s + (0.706 − 0.0136i)14-s + (0.336 + 0.0902i)15-s + (0.125 − 0.216i)16-s + (0.636 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.0139i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.999 + 0.0139i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14492 - 0.00801371i\)
\(L(\frac12)\) \(\approx\) \(1.14492 - 0.00801371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + iT \)
7 \( 1 + (0.635 - 2.56i)T \)
13 \( 1 + (-3.33 + 1.36i)T \)
good5 \( 1 + (0.349 - 1.30i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.88 + 1.88i)T + 11iT^{2} \)
17 \( 1 + (-2.62 - 4.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.88 - 4.88i)T + 19iT^{2} \)
23 \( 1 + (-0.227 - 0.131i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.07 - 5.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.421 - 0.112i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.65 - 0.712i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.81 + 10.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.74 - 3.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.55 + 1.48i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.878 + 1.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.31 + 2.22i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 8.80iT - 61T^{2} \)
67 \( 1 + (7.32 - 7.32i)T - 67iT^{2} \)
71 \( 1 + (-1.08 - 4.04i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.30 - 4.86i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.12 - 5.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.32 + 9.32i)T + 83iT^{2} \)
89 \( 1 + (1.60 + 5.99i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-18.5 + 4.97i)T + (84.0 - 48.5i)T^{2} \)
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

−10.79756693840978782411246096569, −10.16993272704801503625804966467, −8.910059130827811941812674381453, −8.291166990644773664795780997369, −7.37805702063445778562315457096, −6.05431220445149615019954752704, −5.44845353786708976518882301469, −3.55035356876386645559681011730, −2.90042032480628677308508567561, −1.41807995474350667776026190855, 0.811756240341125333046986371000, 3.14366706417594595102046302459, 4.47022952255569380264081246904, 5.00238661431428118112795373387, 6.28964462014808975851273750094, 7.31674418186131013884789481461, 8.026471707936071337003835434373, 9.179463474331745530407642572898, 9.685845802478720643727760444597, 10.64261625365410416538465620656

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