↑

Properties

Label 2-546-21.20-c1-0-30
Degree $2$
Conductor $546$
Sign $-0.965 - 0.259i$
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 + (−0.551 − 1.64i)3-s − 4-s + 0.124·5-s + (−1.64 + 0.551i)6-s + (1.46 − 2.20i)7-s + i·8-s + (−2.39 + 1.81i)9-s − 0.124i·10-s − 3.15i·11-s + (0.551 + 1.64i)12-s − i·13-s + (−2.20 − 1.46i)14-s + (−0.0685 − 0.203i)15-s + 16-s − 5.18·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.318 − 0.947i)3-s − 0.5·4-s + 0.0555·5-s + (−0.670 + 0.225i)6-s + (0.553 − 0.833i)7-s + 0.353i·8-s + (−0.797 + 0.603i)9-s − 0.0392i·10-s − 0.952i·11-s + (0.159 + 0.473i)12-s − 0.277i·13-s + (−0.589 − 0.391i)14-s + (−0.0176 − 0.0526i)15-s + 0.250·16-s − 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.965 - 0.259i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123536 + 0.937178i\)
\(L(\frac12)\) \(\approx\) \(0.123536 + 0.937178i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.551 + 1.64i)T \)
7 \( 1 + (-1.46 + 2.20i)T \)
13 \( 1 + iT \)
good5 \( 1 - 0.124T + 5T^{2} \)
11 \( 1 + 3.15iT - 11T^{2} \)
17 \( 1 + 5.18T + 17T^{2} \)
19 \( 1 + 7.80iT - 19T^{2} \)
23 \( 1 - 8.49iT - 23T^{2} \)
29 \( 1 - 0.988iT - 29T^{2} \)
31 \( 1 - 2.03iT - 31T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 - 0.374T + 41T^{2} \)
43 \( 1 - 0.643T + 43T^{2} \)
47 \( 1 + 0.322T + 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + 6.22T + 59T^{2} \)
61 \( 1 - 9.38iT - 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 - 0.149T + 79T^{2} \)
83 \( 1 + 7.17T + 83T^{2} \)
89 \( 1 - 6.93T + 89T^{2} \)
97 \( 1 - 4.53iT - 97T^{2} \)
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

−10.77874143667833498775367953603, −9.455794095559700381185841027338, −8.524228351911329044662338877311, −7.64936262169869208138577586063, −6.77464433926799332404906925011, −5.61995306777219369491461616432, −4.60366489554237242740667543979, −3.23542883349041292169140546326, −1.91227448200904992527231697508, −0.56325105136758994279651377979, 2.27738712947226338317827721006, 4.08679053356173856020055856871, 4.69945734071581737483891796295, 5.79456001310784139726063989937, 6.46940544489415971940557844637, 7.86172404057987623371484366405, 8.646170574463815851307203756116, 9.476553747764147622252213444111, 10.22032702859471166460996586100, 11.20614225867872026746143096607

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