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Properties

Label 2-546-273.44-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.986 - 0.163i$
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  + (−0.965 + 0.258i)2-s + (0.398 + 1.68i)3-s + (0.866 − 0.499i)4-s + (−1.89 + 0.508i)5-s + (−0.821 − 1.52i)6-s + (1.71 + 2.01i)7-s + (−0.707 + 0.707i)8-s + (−2.68 + 1.34i)9-s + (1.70 − 0.982i)10-s + (2.87 + 0.771i)11-s + (1.18 + 1.26i)12-s + (−3.35 + 1.31i)13-s + (−2.17 − 1.50i)14-s + (−1.61 − 2.99i)15-s + (0.500 − 0.866i)16-s + (1.01 + 1.76i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.230 + 0.973i)3-s + (0.433 − 0.249i)4-s + (−0.848 + 0.227i)5-s + (−0.335 − 0.622i)6-s + (0.647 + 0.761i)7-s + (−0.249 + 0.249i)8-s + (−0.894 + 0.448i)9-s + (0.538 − 0.310i)10-s + (0.867 + 0.232i)11-s + (0.342 + 0.363i)12-s + (−0.930 + 0.365i)13-s + (−0.581 − 0.401i)14-s + (−0.416 − 0.773i)15-s + (0.125 − 0.216i)16-s + (0.247 + 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.986 - 0.163i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0598092 + 0.727106i\)
\(L(\frac12)\) \(\approx\) \(0.0598092 + 0.727106i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.398 - 1.68i)T \)
7 \( 1 + (-1.71 - 2.01i)T \)
13 \( 1 + (3.35 - 1.31i)T \)
good5 \( 1 + (1.89 - 0.508i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.87 - 0.771i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.01 - 1.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.12 + 4.21i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.60 - 2.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.15iT - 29T^{2} \)
31 \( 1 + (2.95 + 0.791i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.92 + 1.05i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.22 + 7.22i)T + 41iT^{2} \)
43 \( 1 + 1.92iT - 43T^{2} \)
47 \( 1 + (-0.203 - 0.758i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.77 - 2.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.58 + 0.692i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.72 + 4.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.655 + 0.175i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.64 - 7.64i)T + 71iT^{2} \)
73 \( 1 + (3.05 - 11.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.89 - 8.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.83 - 4.83i)T + 83iT^{2} \)
89 \( 1 + (3.24 + 12.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.26 + 7.26i)T - 97iT^{2} \)
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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

−11.29443220374423450558015094050, −10.21465714748733470629485860363, −9.291571641044969634750970105762, −8.755116971593713236132385214917, −7.82995494827766125607471752716, −6.95008714869867162673633385009, −5.58627941558207620162560650592, −4.61504612676790137752902992850, −3.51465592344224188095751777572, −2.11779927398228444352938349385, 0.50521510097087000016531924683, 1.83721588347415138621447854478, 3.34955831748384438761634121326, 4.50321850373820080138028938022, 6.09737098474379185296133760008, 7.10770515903781646332845427880, 7.925141622881361757616627407901, 8.213512038754246551955478194578, 9.422393743508178852055948657285, 10.37700814215955734455752154849

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