↑

Properties

Label 2-546-91.5-c1-0-9
Degree $2$
Conductor $546$
Sign $0.717 - 0.697i$
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 + (−0.866 + 0.5i)3-s + (−0.866 + 0.499i)4-s + (1.73 − 0.466i)5-s + (−0.707 − 0.707i)6-s + (2.64 − 0.0195i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.900 + 1.55i)10-s + (0.821 − 3.06i)11-s + (0.5 − 0.866i)12-s + (2.35 − 2.72i)13-s + (0.703 + 2.55i)14-s + (−1.27 + 1.27i)15-s + (0.500 − 0.866i)16-s + (0.111 + 0.193i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 + 0.249i)4-s + (0.777 − 0.208i)5-s + (−0.288 − 0.288i)6-s + (0.999 − 0.00738i)7-s + (−0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.284 + 0.493i)10-s + (0.247 − 0.924i)11-s + (0.144 − 0.249i)12-s + (0.653 − 0.756i)13-s + (0.188 + 0.681i)14-s + (−0.328 + 0.328i)15-s + (0.125 − 0.216i)16-s + (0.0270 + 0.0469i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.717 - 0.697i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.717 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55636 + 0.631807i\)
\(L(\frac12)\) \(\approx\) \(1.55636 + 0.631807i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.64 + 0.0195i)T \)
13 \( 1 + (-2.35 + 2.72i)T \)
good5 \( 1 + (-1.73 + 0.466i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.821 + 3.06i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.111 - 0.193i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.767 + 0.205i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.65 - 1.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.94T + 29T^{2} \)
31 \( 1 + (1.87 - 7.00i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.82 + 1.56i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.12 + 7.12i)T + 41iT^{2} \)
43 \( 1 - 11.7iT - 43T^{2} \)
47 \( 1 + (0.0586 + 0.218i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.0417 - 0.0723i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.62 + 1.23i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.64 - 3.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.81 + 0.753i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.47 - 8.47i)T - 71iT^{2} \)
73 \( 1 + (4.80 + 1.28i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.88 + 5.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.38 + 2.38i)T + 83iT^{2} \)
89 \( 1 + (2.46 + 9.21i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.45 + 8.45i)T + 97iT^{2} \)
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

−10.94427146835382117085584919371, −10.03760219701970866456097898798, −8.952625782992314462098514669638, −8.321088171889030272115323200884, −7.22662188322615879614906761022, −6.01241666755386065976171303605, −5.52876477113275518223922238426, −4.60122909862798271975036017890, −3.27845520504712480098443720254, −1.27376460939608805753094638436, 1.44106124768238831834060102027, 2.32262660490830538004621841892, 4.09727512505728471519383758730, 4.97942194753014198385777866375, 5.98611704543389994902805491592, 6.93064502652145127504445706363, 8.075314491540279716257056665643, 9.167932997072207808147857248762, 9.980418029487388359259237522771, 10.82251476358250241621895834430

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