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Properties

Label 2-546-91.74-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.998 + 0.0496i$
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  − 2-s + (0.5 + 0.866i)3-s + 4-s + (0.114 + 0.197i)5-s + (−0.5 − 0.866i)6-s + (−2.59 − 0.518i)7-s − 8-s + (−0.499 + 0.866i)9-s + (−0.114 − 0.197i)10-s + (−1.70 − 2.95i)11-s + (0.5 + 0.866i)12-s + (−1.62 + 3.21i)13-s + (2.59 + 0.518i)14-s + (−0.114 + 0.197i)15-s + 16-s − 5.41·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (0.0509 + 0.0883i)5-s + (−0.204 − 0.353i)6-s + (−0.980 − 0.195i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.0360 − 0.0624i)10-s + (−0.514 − 0.890i)11-s + (0.144 + 0.249i)12-s + (−0.451 + 0.892i)13-s + (0.693 + 0.138i)14-s + (−0.0294 + 0.0509i)15-s + 0.250·16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.998 + 0.0496i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.998 + 0.0496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00454475 - 0.182871i\)
\(L(\frac12)\) \(\approx\) \(0.00454475 - 0.182871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.59 + 0.518i)T \)
13 \( 1 + (1.62 - 3.21i)T \)
good5 \( 1 + (-0.114 - 0.197i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 + (3.17 - 5.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.91T + 23T^{2} \)
29 \( 1 + (-0.851 + 1.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.78 - 3.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.18T + 37T^{2} \)
41 \( 1 + (-1.59 + 2.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.17 - 8.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.57 + 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.31 - 5.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 14.7T + 59T^{2} \)
61 \( 1 + (-3.35 + 5.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.58 + 9.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.0390 + 0.0677i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.31 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.811 - 1.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.70T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + (-6.82 - 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.81935354027338663121708819243, −10.38086840336612726857610334802, −9.429338184142153738222861434142, −8.766786505149695930710124280115, −7.899133505275936636668818642103, −6.69530119176473762502333724642, −6.04270573226474659643917234093, −4.50965525188392017930640337905, −3.37090577516217940796669472164, −2.20233622014064226469727346574, 0.11533513123380136627020962463, 2.13767372488385520035785765756, 3.04114643256215599596645519179, 4.69894074246873037564268097380, 6.03175126664245836704016289379, 6.96430511633124086026848897806, 7.57225631661544512191651239894, 8.771853754959163635122127584323, 9.298983096389715720131607038633, 10.26519459811440948181175299394

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