Properties

Label 2-546-273.173-c1-0-22
Degree $2$
Conductor $546$
Sign $-0.372 - 0.928i$
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  + (0.5 + 0.866i)2-s + (0.320 + 1.70i)3-s + (−0.499 + 0.866i)4-s + (1.62 + 0.936i)5-s + (−1.31 + 1.12i)6-s + (2.47 − 0.928i)7-s − 0.999·8-s + (−2.79 + 1.08i)9-s + 1.87i·10-s + 5.09·11-s + (−1.63 − 0.573i)12-s + (3.20 − 1.64i)13-s + (2.04 + 1.68i)14-s + (−1.07 + 3.06i)15-s + (−0.5 − 0.866i)16-s + (−1.48 + 2.56i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.184 + 0.982i)3-s + (−0.249 + 0.433i)4-s + (0.725 + 0.418i)5-s + (−0.536 + 0.460i)6-s + (0.936 − 0.350i)7-s − 0.353·8-s + (−0.931 + 0.363i)9-s + 0.592i·10-s + 1.53·11-s + (−0.471 − 0.165i)12-s + (0.889 − 0.456i)13-s + (0.545 + 0.449i)14-s + (−0.277 + 0.790i)15-s + (−0.125 − 0.216i)16-s + (−0.359 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.372 - 0.928i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.372 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24659 + 1.84298i\)
\(L(\frac12)\) \(\approx\) \(1.24659 + 1.84298i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.320 - 1.70i)T \)
7 \( 1 + (-2.47 + 0.928i)T \)
13 \( 1 + (-3.20 + 1.64i)T \)
good5 \( 1 + (-1.62 - 0.936i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 5.09T + 11T^{2} \)
17 \( 1 + (1.48 - 2.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 2.60T + 19T^{2} \)
23 \( 1 + (5.25 - 3.03i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.30 + 4.21i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.09 + 7.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.52 + 3.18i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.85 - 2.80i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.87 + 8.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.70 - 1.56i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.21 - 4.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.90 + 1.10i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 5.90iT - 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 + (-7.27 - 12.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.99 + 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.14 - 8.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.63iT - 83T^{2} \)
89 \( 1 + (-12.7 + 7.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.83 + 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.11877387898562288205664550617, −10.11124567508984114474612479684, −9.270303593875117938749454933342, −8.459203463539079333766316313765, −7.55149041478972032147852394933, −6.09762805870445736753488775793, −5.78284348085469990989442840311, −4.18501580103901002075761536795, −3.87593645358775296255209857149, −2.04209459794585230446882930578, 1.42499978070207048756018914816, 1.99224025325589376840773272718, 3.62479514201048453371225026725, 4.86044382038371304567088817348, 5.99043828218718572054181738910, 6.63191803636985034299935971153, 7.992334502307415407905970321571, 9.027049439060753514751239916849, 9.286224423303540816216501750979, 10.93719594885047786622211472378

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