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Properties

Label 2-546-273.185-c1-0-26
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: $\chi_{546} (185, \cdot )$
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

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

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