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Properties

Label 2-546-91.24-c1-0-9
Degree $2$
Conductor $546$
Sign $0.485 + 0.874i$
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 + i·3-s + (0.866 − 0.499i)4-s + (0.195 + 0.0524i)5-s + (−0.258 − 0.965i)6-s + (−2.35 − 1.21i)7-s + (−0.707 + 0.707i)8-s − 9-s − 0.202·10-s + (1.97 − 1.97i)11-s + (0.499 + 0.866i)12-s + (−1.90 − 3.06i)13-s + (2.58 + 0.564i)14-s + (−0.0524 + 0.195i)15-s + (0.500 − 0.866i)16-s + (−1.34 − 2.32i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (0.0874 + 0.0234i)5-s + (−0.105 − 0.394i)6-s + (−0.888 − 0.458i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s − 0.0640·10-s + (0.595 − 0.595i)11-s + (0.144 + 0.249i)12-s + (−0.528 − 0.848i)13-s + (0.690 + 0.150i)14-s + (−0.0135 + 0.0505i)15-s + (0.125 − 0.216i)16-s + (−0.326 − 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.485 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632607 - 0.372171i\)
\(L(\frac12)\) \(\approx\) \(0.632607 - 0.372171i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (2.35 + 1.21i)T \)
13 \( 1 + (1.90 + 3.06i)T \)
good5 \( 1 + (-0.195 - 0.0524i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.97 + 1.97i)T - 11iT^{2} \)
17 \( 1 + (1.34 + 2.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.84 + 2.84i)T - 19iT^{2} \)
23 \( 1 + (-1.97 - 1.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.82 - 3.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.51 + 5.67i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.427 + 1.59i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-10.6 - 2.85i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.73 + 3.31i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.56 + 5.83i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.577 + 1.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.27 + 4.76i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 2.01iT - 61T^{2} \)
67 \( 1 + (7.22 + 7.22i)T + 67iT^{2} \)
71 \( 1 + (-1.05 + 0.283i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-11.4 + 3.07i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0814 + 0.141i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.03 + 5.03i)T - 83iT^{2} \)
89 \( 1 + (14.6 - 3.91i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.70 - 10.0i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.50075087372165247513152425234, −9.582862114611055470205628629518, −9.223526815465708820129361918107, −8.023085519101450451756012002582, −7.10105313130140265816112792399, −6.17608360411107970077053002333, −5.15981112682515276179276544495, −3.75674256960452495896964661484, −2.70548965879266223680470421305, −0.53490149646482999038897076010, 1.56550515875459935037715181651, 2.74486376943710025890686511186, 4.09667767679106341403508167549, 5.70596918467397041575864014774, 6.63573961119354986518480535361, 7.29416259240267349740652901380, 8.388799202973387701214878662310, 9.353227127993765164234574703712, 9.767322490524708544080422304628, 10.93604594310287382731676486054

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