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Properties

Label 2-546-21.17-c1-0-13
Degree $2$
Conductor $546$
Sign $0.753 + 0.657i$
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 + (−1.73 + 0.0542i)3-s + (0.499 − 0.866i)4-s + (−0.401 − 0.695i)5-s + (−1.47 + 0.912i)6-s + (1.69 + 2.03i)7-s − 0.999i·8-s + (2.99 − 0.187i)9-s + (−0.695 − 0.401i)10-s + (1.31 + 0.760i)11-s + (−0.818 + 1.52i)12-s + i·13-s + (2.48 + 0.913i)14-s + (0.732 + 1.18i)15-s + (−0.5 − 0.866i)16-s + (2.74 − 4.76i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.999 + 0.0313i)3-s + (0.249 − 0.433i)4-s + (−0.179 − 0.311i)5-s + (−0.600 + 0.372i)6-s + (0.640 + 0.768i)7-s − 0.353i·8-s + (0.998 − 0.0626i)9-s + (−0.219 − 0.126i)10-s + (0.397 + 0.229i)11-s + (−0.236 + 0.440i)12-s + 0.277i·13-s + (0.663 + 0.244i)14-s + (0.189 + 0.305i)15-s + (−0.125 − 0.216i)16-s + (0.666 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53192 - 0.574848i\)
\(L(\frac12)\) \(\approx\) \(1.53192 - 0.574848i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 0.0542i)T \)
7 \( 1 + (-1.69 - 2.03i)T \)
13 \( 1 - iT \)
good5 \( 1 + (0.401 + 0.695i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.31 - 0.760i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.74 + 4.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.336 - 0.194i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.51 + 2.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.22iT - 29T^{2} \)
31 \( 1 + (-2.61 - 1.50i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.08 - 8.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.60T + 41T^{2} \)
43 \( 1 - 7.00T + 43T^{2} \)
47 \( 1 + (-0.663 - 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.21 + 3.58i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.198 + 0.344i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.18 - 4.72i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.134 - 0.232i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.19iT - 71T^{2} \)
73 \( 1 + (13.1 + 7.60i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.57 - 9.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.35T + 83T^{2} \)
89 \( 1 + (-7.89 - 13.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.73iT - 97T^{2} \)
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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

−11.01540937619368855861606658038, −9.996819438890196071290219877716, −9.149477747479889025176550332629, −7.926799091046375784470181510308, −6.78629971537136721063259649781, −5.91209918571542854803423002759, −4.90443358344564180676681719915, −4.40990124440956233636021438642, −2.70195662872379166284561270342, −1.13528927532026746817941940429, 1.34210860019855360969551434513, 3.43579136549517828936237136781, 4.39037702425854244217280931306, 5.36562159452905055447985744299, 6.22383630141440729336891728812, 7.22381330564289448488020529405, 7.77839417006243454030732449037, 9.130334608577798689810080867712, 10.52178223568709677559700289144, 10.91085598188045081464869527873

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