Properties

Label 2-546-13.12-c1-0-11
Degree $2$
Conductor $546$
Sign $-0.987 - 0.155i$
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  i·2-s − 3-s − 4-s − 3.56i·5-s + i·6-s + i·7-s + i·8-s + 9-s − 3.56·10-s − 5.56i·11-s + 12-s + (3.56 + 0.561i)13-s + 14-s + 3.56i·15-s + 16-s − 6.68·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 1.59i·5-s + 0.408i·6-s + 0.377i·7-s + 0.353i·8-s + 0.333·9-s − 1.12·10-s − 1.67i·11-s + 0.288·12-s + (0.987 + 0.155i)13-s + 0.267·14-s + 0.919i·15-s + 0.250·16-s − 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.987 - 0.155i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0603192 + 0.769855i\)
\(L(\frac12)\) \(\approx\) \(0.0603192 + 0.769855i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + T \)
7 \( 1 - iT \)
13 \( 1 + (-3.56 - 0.561i)T \)
good5 \( 1 + 3.56iT - 5T^{2} \)
11 \( 1 + 5.56iT - 11T^{2} \)
17 \( 1 + 6.68T + 17T^{2} \)
19 \( 1 - 1.56iT - 19T^{2} \)
23 \( 1 + 6.68T + 23T^{2} \)
29 \( 1 - 1.56T + 29T^{2} \)
31 \( 1 - 6.24iT - 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + 6.43T + 43T^{2} \)
47 \( 1 - 10.2iT - 47T^{2} \)
53 \( 1 - 4.87T + 53T^{2} \)
59 \( 1 + 4.24iT - 59T^{2} \)
61 \( 1 + 1.56T + 61T^{2} \)
67 \( 1 - 1.12iT - 67T^{2} \)
71 \( 1 + 9.36iT - 71T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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

−10.63773064596333522986362743562, −9.268648105364263272621926553473, −8.742248087836666161869673482762, −8.131688311889702758146594966132, −6.27230941465201258216522956180, −5.60423596972005402576376875847, −4.58406684446153368875205253239, −3.64149204114418432137774574141, −1.80185961144397048832004952143, −0.47871636239499742434539236738, 2.19407707512459937400723072633, 3.82052227468415415111274129880, 4.71037456134547220526955053946, 6.17339577438278904287078365950, 6.70208531573655843371257144585, 7.32072216481566907410537520651, 8.379764983294535442060579462775, 9.817858753554088330932664315798, 10.25546804326659356622495519310, 11.19221015885123806942260079548

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