Properties

Label 2-546-91.25-c1-0-9
Degree $2$
Conductor $546$
Sign $0.993 + 0.116i$
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 + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (2.62 − 1.51i)5-s + 0.999i·6-s + (2.43 + 1.04i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−1.51 + 2.62i)10-s + (3.75 + 2.16i)11-s + (−0.499 − 0.866i)12-s + (−0.613 + 3.55i)13-s + (−2.62 + 0.311i)14-s − 3.02i·15-s + (−0.5 − 0.866i)16-s + (−0.721 + 1.25i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (1.17 − 0.676i)5-s + 0.408i·6-s + (0.918 + 0.394i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (−0.478 + 0.829i)10-s + (1.13 + 0.653i)11-s + (−0.144 − 0.249i)12-s + (−0.170 + 0.985i)13-s + (−0.702 + 0.0831i)14-s − 0.781i·15-s + (−0.125 − 0.216i)16-s + (−0.175 + 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62358 - 0.0946433i\)
\(L(\frac12)\) \(\approx\) \(1.62358 - 0.0946433i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.43 - 1.04i)T \)
13 \( 1 + (0.613 - 3.55i)T \)
good5 \( 1 + (-2.62 + 1.51i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.75 - 2.16i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.721 - 1.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.12 - 0.652i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.42T + 29T^{2} \)
31 \( 1 + (1.46 + 0.847i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.89 - 5.71i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.53iT - 41T^{2} \)
43 \( 1 + 5.10T + 43T^{2} \)
47 \( 1 + (-7.28 + 4.20i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.05 + 3.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.34 + 2.50i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.505 + 0.875i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.412 + 0.238i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (-3.42 - 1.97i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.34 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.0iT - 83T^{2} \)
89 \( 1 + (6.04 - 3.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
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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

−10.54001952005028192398212562514, −9.645961945600721796363419894168, −8.828371974463917002297331341674, −8.475662219865134465280059265819, −7.10525044739723835198583494356, −6.40278314927122059622499992687, −5.38065433926114441705173067348, −4.32185577163103912425546168307, −2.07271458455986985916146775927, −1.56315060232583956956166894545, 1.47946837832586067353110970447, 2.73388509126172457627369247109, 3.86056257085030624195789561496, 5.26049102870030624238794636799, 6.29562400325052327354302221524, 7.36423498104698312915391910199, 8.378273190003250996803411270657, 9.195393340642317290656572398414, 10.00346618028256843841300432915, 10.67690136887016786371211641441

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