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Properties

Label 2-546-91.88-c1-0-10
Degree $2$
Conductor $546$
Sign $-0.426 + 0.904i$
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  + (−0.866 − 0.5i)2-s + 3-s + (0.499 + 0.866i)4-s + (−2.87 + 1.65i)5-s + (−0.866 − 0.5i)6-s + (−2.57 + 0.624i)7-s − 0.999i·8-s + 9-s + 3.31·10-s − 4.84i·11-s + (0.499 + 0.866i)12-s + (2.96 + 2.04i)13-s + (2.53 + 0.744i)14-s + (−2.87 + 1.65i)15-s + (−0.5 + 0.866i)16-s + (−2.71 − 4.69i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + 0.577·3-s + (0.249 + 0.433i)4-s + (−1.28 + 0.742i)5-s + (−0.353 − 0.204i)6-s + (−0.971 + 0.236i)7-s − 0.353i·8-s + 0.333·9-s + 1.04·10-s − 1.45i·11-s + (0.144 + 0.249i)12-s + (0.823 + 0.567i)13-s + (0.678 + 0.198i)14-s + (−0.742 + 0.428i)15-s + (−0.125 + 0.216i)16-s + (−0.657 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.426 + 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308953 - 0.487387i\)
\(L(\frac12)\) \(\approx\) \(0.308953 - 0.487387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 - T \)
7 \( 1 + (2.57 - 0.624i)T \)
13 \( 1 + (-2.96 - 2.04i)T \)
good5 \( 1 + (2.87 - 1.65i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 4.84iT - 11T^{2} \)
17 \( 1 + (2.71 + 4.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 6.97iT - 19T^{2} \)
23 \( 1 + (-3.11 + 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.459 + 0.795i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.25 + 2.45i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.76 - 3.90i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.91 - 3.98i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.86 - 4.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.99 - 1.15i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.54 + 4.40i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.30 - 4.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 3.64T + 61T^{2} \)
67 \( 1 + 8.80iT - 67T^{2} \)
71 \( 1 + (6.84 + 3.95i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.61 + 0.929i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.39 + 2.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.97iT - 83T^{2} \)
89 \( 1 + (-5.38 - 3.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.466 + 0.269i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.74818816722653447329164504556, −9.375764113092895220262678335676, −8.851172731172979418452405040068, −8.029094409455888202645295155263, −6.95468848690128734974372324661, −6.42010238122889523668719466888, −4.43882378172888403065595087224, −3.23119569715586589937526816678, −2.84196065156926568608333732635, −0.38847089951782710426561557909, 1.56615056973684940246676082621, 3.51225270627516519073232924117, 4.20101695525078231963784221074, 5.65418638376301516885015601553, 6.92599568669339515713828040036, 7.64904571068435227972811667298, 8.382328573575732381374174116370, 9.156292777593347344574707518240, 10.04241460441358928375284130545, 10.83254375647814474314232160172

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