Properties

Label 2-546-273.152-c1-0-23
Degree $2$
Conductor $546$
Sign $0.000615 + 0.999i$
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 + (−0.219 + 1.71i)3-s + (0.499 − 0.866i)4-s + (0.356 − 0.616i)5-s + (−0.668 − 1.59i)6-s + (−2.23 + 1.42i)7-s + 0.999i·8-s + (−2.90 − 0.754i)9-s + 0.712i·10-s − 5.62i·11-s + (1.37 + 1.04i)12-s + (−3.20 + 1.66i)13-s + (1.22 − 2.34i)14-s + (0.981 + 0.747i)15-s + (−0.5 − 0.866i)16-s + (0.965 − 1.67i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.126 + 0.991i)3-s + (0.249 − 0.433i)4-s + (0.159 − 0.275i)5-s + (−0.273 − 0.652i)6-s + (−0.843 + 0.537i)7-s + 0.353i·8-s + (−0.967 − 0.251i)9-s + 0.225i·10-s − 1.69i·11-s + (0.397 + 0.302i)12-s + (−0.887 + 0.460i)13-s + (0.326 − 0.627i)14-s + (0.253 + 0.192i)15-s + (−0.125 − 0.216i)16-s + (0.234 − 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000615 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.000615 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.000615 + 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.000615 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182445 - 0.182333i\)
\(L(\frac12)\) \(\approx\) \(0.182445 - 0.182333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.219 - 1.71i)T \)
7 \( 1 + (2.23 - 1.42i)T \)
13 \( 1 + (3.20 - 1.66i)T \)
good5 \( 1 + (-0.356 + 0.616i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.62iT - 11T^{2} \)
17 \( 1 + (-0.965 + 1.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 5.54iT - 19T^{2} \)
23 \( 1 + (6.23 - 3.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.73 + 4.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.0986 + 0.0569i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.54 - 4.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.05 + 1.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.02 + 6.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.78 + 4.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.94 + 1.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.64 + 4.58i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 5.75iT - 61T^{2} \)
67 \( 1 + 3.54T + 67T^{2} \)
71 \( 1 + (5.97 - 3.44i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.54 + 4.93i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.594 + 1.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + (-5.78 - 10.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.57 - 4.37i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35171272321420328284205981036, −9.429951235827068847066379998131, −9.138555614914925762587662060901, −8.186227273360726205948865593444, −6.89755887542786642739610590022, −5.80207662528156018090491494388, −5.28740528654612487310102325139, −3.76661334135926314675298852169, −2.62684975196725562649570397571, −0.17364679899648848331562211451, 1.71259476094048013129062886983, 2.75928502202776903559680169401, 4.18588325339951881991638499986, 5.79737632811100540039369898081, 6.76813729660531412244460836225, 7.45912620702462363811554704663, 8.128081730592157327404815006609, 9.532306421041595888841161249703, 10.08386259118932163567904235214, 10.82619425090080513341081700825

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