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Properties

Label 2-546-273.173-c1-0-14
Degree $2$
Conductor $546$
Sign $0.955 + 0.293i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−1.71 − 0.243i)3-s + (−0.499 + 0.866i)4-s + (−2.87 − 1.66i)5-s + (−0.646 − 1.60i)6-s + (−0.187 + 2.63i)7-s − 0.999·8-s + (2.88 + 0.834i)9-s − 3.32i·10-s + 1.48·11-s + (1.06 − 1.36i)12-s + (−1.88 − 3.07i)13-s + (−2.37 + 1.15i)14-s + (4.53 + 3.55i)15-s + (−0.5 − 0.866i)16-s + (2.81 − 4.88i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.990 − 0.140i)3-s + (−0.249 + 0.433i)4-s + (−1.28 − 0.743i)5-s + (−0.263 − 0.655i)6-s + (−0.0707 + 0.997i)7-s − 0.353·8-s + (0.960 + 0.278i)9-s − 1.05i·10-s + 0.447·11-s + (0.308 − 0.393i)12-s + (−0.523 − 0.851i)13-s + (−0.635 + 0.309i)14-s + (1.17 + 0.917i)15-s + (−0.125 − 0.216i)16-s + (0.683 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.955 + 0.293i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.955 + 0.293i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851383 - 0.127728i\)
\(L(\frac12)\) \(\approx\) \(0.851383 - 0.127728i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.243i)T \)
7 \( 1 + (0.187 - 2.63i)T \)
13 \( 1 + (1.88 + 3.07i)T \)
good5 \( 1 + (2.87 + 1.66i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.48T + 11T^{2} \)
17 \( 1 + (-2.81 + 4.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 5.36T + 19T^{2} \)
23 \( 1 + (-3.47 + 2.00i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.127 + 0.0736i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.689 + 1.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.80 + 5.08i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.728 - 0.420i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.56 + 7.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.41 - 4.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.6 - 6.15i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.131 - 0.0757i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.30iT - 61T^{2} \)
67 \( 1 + 9.94iT - 67T^{2} \)
71 \( 1 + (-2.25 - 3.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.99 + 3.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.75 + 3.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + (1.49 - 0.863i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.27 + 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.17301775498708817208310753988, −9.717571989988605194949677581378, −8.897947865347073225499290976509, −7.69991118160078412410584970299, −7.33170025742401117454762435224, −5.94116017816739630225025218911, −5.19791427364225125296937376496, −4.48902815079466571479318265469, −3.10946262031914239457282921165, −0.63515919975680213165025441597, 1.16099417267466572626038653038, 3.37308278204362084083332203587, 4.03657818349657474755322562536, 4.94291510218423947392706885506, 6.33940717952597903572959588966, 7.12141973532990265839382826261, 7.891780014012159286467634900344, 9.552251885603997712365134129956, 10.21072268966092841403119637334, 11.22955931596309862110022362728

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