Properties

Label 2-546-273.17-c1-0-23
Degree $2$
Conductor $546$
Sign $0.928 - 0.370i$
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  + 2-s + (1.06 + 1.36i)3-s + 4-s + (2.87 − 1.66i)5-s + (1.06 + 1.36i)6-s + (−2.37 − 1.15i)7-s + 8-s + (−0.717 + 2.91i)9-s + (2.87 − 1.66i)10-s + (0.741 + 1.28i)11-s + (1.06 + 1.36i)12-s + (1.88 + 3.07i)13-s + (−2.37 − 1.15i)14-s + (5.34 + 2.15i)15-s + 16-s − 5.63·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.616 + 0.787i)3-s + 0.5·4-s + (1.28 − 0.743i)5-s + (0.436 + 0.556i)6-s + (−0.899 − 0.437i)7-s + 0.353·8-s + (−0.239 + 0.970i)9-s + (0.910 − 0.525i)10-s + (0.223 + 0.387i)11-s + (0.308 + 0.393i)12-s + (0.523 + 0.851i)13-s + (−0.635 − 0.309i)14-s + (1.37 + 0.555i)15-s + 0.250·16-s − 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.928 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90609 + 0.557815i\)
\(L(\frac12)\) \(\approx\) \(2.90609 + 0.557815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.06 - 1.36i)T \)
7 \( 1 + (2.37 + 1.15i)T \)
13 \( 1 + (-1.88 - 3.07i)T \)
good5 \( 1 + (-2.87 + 1.66i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.741 - 1.28i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.63T + 17T^{2} \)
19 \( 1 + (-2.68 + 4.64i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + (-0.127 - 0.0736i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.689 + 1.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 + (-0.728 - 0.420i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.56 + 7.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.41 - 4.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.6 + 6.15i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.151iT - 59T^{2} \)
61 \( 1 + (-6.32 - 3.65i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.61 - 4.97i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.25 + 3.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.99 + 3.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.75 - 3.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + 1.72iT - 89T^{2} \)
97 \( 1 + (-7.27 - 12.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.72061567918730818251558629479, −9.819156365712694702124132887773, −9.277922764508736707928237342385, −8.516824707109536452058846994961, −6.89732570526257048377757831273, −6.23628500746992760618426975618, −4.93520843599846000964328488552, −4.36570626581875595381338482377, −3.05362784814254260322371611894, −1.89351218600444645906290304129, 1.77567972313766057761027995796, 2.84510232724050464633074702069, 3.57114395076344663790692947840, 5.57553248215397458783366539502, 6.19349350884497442778119107019, 6.77615137764741188564664899571, 7.932750331461894090218954929843, 9.097153514799985519345328517679, 9.797557199964470652223290682107, 10.79239685854729244631298180960

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