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Properties

Label 2-546-273.251-c1-0-25
Degree $2$
Conductor $546$
Sign $0.430 + 0.902i$
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 + (1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 3.46i·5-s + (1.5 − 0.866i)6-s + (2.5 + 0.866i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−2.99 − 1.73i)10-s + (−1.5 + 2.59i)11-s − 1.73i·12-s + (3.5 − 0.866i)13-s + (2 − 1.73i)14-s + (2.99 − 5.19i)15-s + (−0.5 + 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 1.54i·5-s + (0.612 − 0.353i)6-s + (0.944 + 0.327i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.948 − 0.547i)10-s + (−0.452 + 0.783i)11-s − 0.499i·12-s + (0.970 − 0.240i)13-s + (0.534 − 0.462i)14-s + (0.774 − 1.34i)15-s + (−0.125 + 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.430 + 0.902i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.430 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03406 - 1.28275i\)
\(L(\frac12)\) \(\approx\) \(2.03406 - 1.28275i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6 - 3.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.66iT - 47T^{2} \)
53 \( 1 - 8.66iT - 53T^{2} \)
59 \( 1 + (-9 + 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (7.5 + 4.33i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.82318833695876405541139087297, −9.507122661028961176946428581338, −8.954642786528935692383632500401, −8.399518132699041492276816647180, −7.29334407394395084745429439575, −5.35723241514589119346424560334, −4.85466274263137514455950931304, −4.07531622514595584239566222180, −2.55794422222255809473022854115, −1.40282037575030791985896656203, 1.94506129643303569782553429207, 3.28623842475651424666593322353, 3.98626857949496222011960525997, 5.67233096032637178991782036792, 6.66808747737944856848212619749, 7.20487540778183537262662955882, 8.358243979180356856401867568068, 8.583335330917070372761100024260, 10.30765734111193944929865928702, 10.84177431309224925357041016537

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