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Properties

Label 2-546-273.152-c1-0-12
Degree $2$
Conductor $546$
Sign $0.375 - 0.926i$
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 + (1.72 + 0.195i)3-s + (0.499 − 0.866i)4-s + (0.349 − 0.605i)5-s + (−1.58 + 0.691i)6-s + (−1.78 + 1.95i)7-s + 0.999i·8-s + (2.92 + 0.671i)9-s + 0.698i·10-s + 1.47i·11-s + (1.02 − 1.39i)12-s + (−0.668 + 3.54i)13-s + (0.563 − 2.58i)14-s + (0.719 − 0.973i)15-s + (−0.5 − 0.866i)16-s + (0.789 − 1.36i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.993 + 0.112i)3-s + (0.249 − 0.433i)4-s + (0.156 − 0.270i)5-s + (−0.648 + 0.282i)6-s + (−0.672 + 0.739i)7-s + 0.353i·8-s + (0.974 + 0.223i)9-s + 0.221i·10-s + 0.445i·11-s + (0.297 − 0.402i)12-s + (−0.185 + 0.982i)13-s + (0.150 − 0.690i)14-s + (0.185 − 0.251i)15-s + (−0.125 − 0.216i)16-s + (0.191 − 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.375 - 0.926i$
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: \((2,\ 546,\ (\ :1/2),\ 0.375 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21213 + 0.816564i\)
\(L(\frac12)\) \(\approx\) \(1.21213 + 0.816564i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.195i)T \)
7 \( 1 + (1.78 - 1.95i)T \)
13 \( 1 + (0.668 - 3.54i)T \)
good5 \( 1 + (-0.349 + 0.605i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.47iT - 11T^{2} \)
17 \( 1 + (-0.789 + 1.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 7.68iT - 19T^{2} \)
23 \( 1 + (-6.95 + 4.01i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.06 + 2.34i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.83 + 3.36i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.914 + 1.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.63 + 2.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.35 + 2.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.64 + 6.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.97 - 4.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.89 - 11.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 3.69iT - 61T^{2} \)
67 \( 1 + 3.95T + 67T^{2} \)
71 \( 1 + (-11.6 + 6.73i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (12.7 - 7.35i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.64 - 4.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.659T + 83T^{2} \)
89 \( 1 + (3.02 + 5.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.8 + 6.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.59699146495626381660401815402, −9.692417767204377857661685207534, −9.210960195727686111977371162440, −8.506335099883723747052790657672, −7.48773188984976599975878690637, −6.67326217998384013828928289848, −5.52538928002213630132625144769, −4.25818980759637195848807008000, −2.89214467841880252953061330432, −1.72448929117208277739747690101, 1.00484386898046584713696501015, 2.82735183787744683792052004536, 3.29422778339505860904900160988, 4.75489427191322557874083894831, 6.47857070383763131865536691697, 7.21345810435253088861067968723, 8.042766155825198895962977380368, 9.007845982742414010646495075596, 9.655540385981635122840305276741, 10.53002795222949357469912558096

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