↑

Properties

Label 2-546-39.32-c1-0-15
Degree $2$
Conductor $546$
Sign $0.317 + 0.948i$
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 + (0.523 + 1.65i)3-s + (0.866 − 0.499i)4-s + (−1.11 − 1.11i)5-s + (−0.932 − 1.45i)6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (−2.45 + 1.72i)9-s + (1.36 + 0.785i)10-s + (−1.03 − 3.86i)11-s + (1.27 + 1.16i)12-s + (−1.97 − 3.01i)13-s + i·14-s + (1.25 − 2.41i)15-s + (0.500 − 0.866i)16-s + (−2.65 − 4.59i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.301 + 0.953i)3-s + (0.433 − 0.249i)4-s + (−0.496 − 0.496i)5-s + (−0.380 − 0.595i)6-s + (0.0978 − 0.365i)7-s + (−0.249 + 0.249i)8-s + (−0.817 + 0.575i)9-s + (0.430 + 0.248i)10-s + (−0.311 − 1.16i)11-s + (0.369 + 0.337i)12-s + (−0.546 − 0.837i)13-s + 0.267i·14-s + (0.323 − 0.623i)15-s + (0.125 − 0.216i)16-s + (−0.643 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.317 + 0.948i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539009 - 0.388091i\)
\(L(\frac12)\) \(\approx\) \(0.539009 - 0.388091i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.523 - 1.65i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (1.97 + 3.01i)T \)
good5 \( 1 + (1.11 + 1.11i)T + 5iT^{2} \)
11 \( 1 + (1.03 + 3.86i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.65 + 4.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.14 - 1.37i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.44 - 2.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.12 + 4.11i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.52 - 3.52i)T - 31iT^{2} \)
37 \( 1 + (-9.27 + 2.48i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.976 + 0.261i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.645 - 0.372i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.72 + 4.72i)T - 47iT^{2} \)
53 \( 1 - 7.57iT - 53T^{2} \)
59 \( 1 + (-9.73 - 2.60i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.42 - 2.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.21 + 15.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.21 + 11.9i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.67 + 3.67i)T + 73iT^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + (2.04 + 2.04i)T + 83iT^{2} \)
89 \( 1 + (-4.10 - 15.3i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.62 + 2.04i)T + (84.0 + 48.5i)T^{2} \)
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

−10.53657926591146038003396495580, −9.586861314072239006617329939060, −8.993488358508340791456079638321, −7.953338300320203183138995527380, −7.52615062303722823282766330583, −5.82616278156201696561171600803, −5.06267968663296035582076190982, −3.81821733642727647488046242952, −2.71409698645177319683231399628, −0.44730939568357125007048710508, 1.74525788533850308563610315646, 2.67579935854228889458264571452, 4.07314298490936489591518363069, 5.68081924025636774105181213073, 6.97923220940025669220997708968, 7.29925185395461976638374958345, 8.232233353357343757245381535666, 9.181284360741163284812433439356, 9.916867514669621303617332545544, 11.27536477288017337778490819742

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