↑

Properties

Label 2-546-7.2-c1-0-10
Degree $2$
Conductor $546$
Sign $0.601 + 0.798i$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.70 − 2.96i)5-s + 0.999·6-s + (2.36 + 1.18i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.70 − 2.96i)10-s + (2.36 + 4.09i)11-s + (0.499 − 0.866i)12-s + 13-s + (2.20 − 1.45i)14-s + 3.41·15-s + (−0.5 + 0.866i)16-s + (−3.89 − 6.75i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.764 − 1.32i)5-s + 0.408·6-s + (0.893 + 0.448i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.540 − 0.936i)10-s + (0.713 + 1.23i)11-s + (0.144 − 0.249i)12-s + 0.277·13-s + (0.590 − 0.388i)14-s + 0.882·15-s + (−0.125 + 0.216i)16-s + (−0.945 − 1.63i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.601 + 0.798i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.601 + 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02963 - 1.01178i\)
\(L(\frac12)\) \(\approx\) \(2.02963 - 1.01178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.36 - 1.18i)T \)
13 \( 1 - T \)
good5 \( 1 + (-1.70 + 2.96i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.89 + 6.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.34 - 2.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.86 + 3.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-2.84 - 4.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.55 + 2.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.892T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + (-5.07 + 8.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.60 - 7.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.05 - 5.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.52 - 2.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0326 - 0.0565i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + (5.86 + 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.26 - 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.934T + 83T^{2} \)
89 \( 1 + (7.55 - 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.79T + 97T^{2} \)
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

−10.61816861896486938717305582032, −9.686049282194869836251609118622, −9.024304872424472873441885714529, −8.520159214544773279270949361932, −7.00318799469975665559464748223, −5.57246334176930919671483599473, −4.80509587627537320222355586491, −4.28028703095925794001504329419, −2.45018421628025211418214988604, −1.46686257386554391633459744672, 1.76219057436260182521468175612, 3.14270063176732897899104736916, 4.18454020951522585374476137090, 5.78552701399248893410569427022, 6.38933595465038270511678833949, 7.11596041864732695219158024912, 8.190400324460149184752342662956, 8.847930854676286298927251484553, 10.13607939070876663595691970951, 11.14769474177910737440391266792

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