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Properties

Label 2-546-91.88-c1-0-8
Degree $2$
Conductor $546$
Sign $0.976 + 0.216i$
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 + 3-s + (0.499 + 0.866i)4-s + (2.59 − 1.49i)5-s + (−0.866 − 0.5i)6-s + (0.521 + 2.59i)7-s − 0.999i·8-s + 9-s − 2.99·10-s + 0.776i·11-s + (0.499 + 0.866i)12-s + (3.39 + 1.20i)13-s + (0.844 − 2.50i)14-s + (2.59 − 1.49i)15-s + (−0.5 + 0.866i)16-s + (−1.17 − 2.02i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + 0.577·3-s + (0.249 + 0.433i)4-s + (1.16 − 0.670i)5-s + (−0.353 − 0.204i)6-s + (0.197 + 0.980i)7-s − 0.353i·8-s + 0.333·9-s − 0.947·10-s + 0.234i·11-s + (0.144 + 0.249i)12-s + (0.942 + 0.334i)13-s + (0.225 − 0.670i)14-s + (0.670 − 0.386i)15-s + (−0.125 + 0.216i)16-s + (−0.283 − 0.491i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.976 + 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65324 - 0.181357i\)
\(L(\frac12)\) \(\approx\) \(1.65324 - 0.181357i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (-0.521 - 2.59i)T \)
13 \( 1 + (-3.39 - 1.20i)T \)
good5 \( 1 + (-2.59 + 1.49i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.776iT - 11T^{2} \)
17 \( 1 + (1.17 + 2.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 2.95iT - 19T^{2} \)
23 \( 1 + (1.03 - 1.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.541 - 0.937i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.31 + 3.64i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.95 + 3.43i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.81 + 5.66i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.64 - 4.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.35 + 4.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.30 + 3.99i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.28 - 1.89i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 5.34iT - 67T^{2} \)
71 \( 1 + (3.56 + 2.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.95 + 4.01i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.49 + 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.25iT - 83T^{2} \)
89 \( 1 + (7.54 + 4.35i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.45 - 1.41i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.58662825002834050644307379140, −9.582882436341055458055378689222, −9.078060802695564033708593441087, −8.523412610708002911719115517047, −7.40366776875236421509196521805, −6.10779524246440035558110173756, −5.30971881487198578508026485294, −3.84462886474665258609690963835, −2.36638835935719503577474485323, −1.59020415975853310347097451958, 1.38253597132440212092690698456, 2.69286475735590211715069604840, 4.02056681437136468576611832017, 5.53897904859723975655072834789, 6.50141613471240904626068589567, 7.19124456339385353700811905507, 8.259742354750607212999137737318, 9.031120513342361531445747326158, 9.986827871616667119595017546405, 10.60795363746510419825923152384

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