Properties

Label 2-546-273.257-c1-0-25
Degree $2$
Conductor $546$
Sign $0.737 + 0.675i$
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 + (−0.889 + 1.48i)3-s + 4-s + (−2.84 − 1.64i)5-s + (−0.889 + 1.48i)6-s + (1.87 − 1.86i)7-s + 8-s + (−1.41 − 2.64i)9-s + (−2.84 − 1.64i)10-s + (1.03 − 1.79i)11-s + (−0.889 + 1.48i)12-s + (3.53 − 0.716i)13-s + (1.87 − 1.86i)14-s + (4.97 − 2.76i)15-s + 16-s − 1.12·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.513 + 0.857i)3-s + 0.5·4-s + (−1.27 − 0.735i)5-s + (−0.363 + 0.606i)6-s + (0.707 − 0.706i)7-s + 0.353·8-s + (−0.472 − 0.881i)9-s + (−0.900 − 0.519i)10-s + (0.312 − 0.540i)11-s + (−0.256 + 0.428i)12-s + (0.980 − 0.198i)13-s + (0.500 − 0.499i)14-s + (1.28 − 0.714i)15-s + 0.250·16-s − 0.272·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45130 - 0.564195i\)
\(L(\frac12)\) \(\approx\) \(1.45130 - 0.564195i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.889 - 1.48i)T \)
7 \( 1 + (-1.87 + 1.86i)T \)
13 \( 1 + (-3.53 + 0.716i)T \)
good5 \( 1 + (2.84 + 1.64i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.03 + 1.79i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 + (0.505 + 0.875i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.50iT - 23T^{2} \)
29 \( 1 + (-7.97 + 4.60i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.86 + 3.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.97iT - 37T^{2} \)
41 \( 1 + (8.52 - 4.92i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.35 + 5.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.05 + 0.609i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.05 + 2.92i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.80iT - 59T^{2} \)
61 \( 1 + (0.209 - 0.120i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.2 + 5.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.94 - 6.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.878 - 1.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.48 - 4.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.999iT - 83T^{2} \)
89 \( 1 + 7.61iT - 89T^{2} \)
97 \( 1 + (3.21 - 5.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.93862610283636473826172735882, −10.16301137497585451637537245168, −8.626758351022788702736895699453, −8.250166437233154667364937602130, −6.91518126081158773804336291533, −5.86685148679346541287367529293, −4.64601999066596619304303481666, −4.27409134124194041065448046144, −3.33806953177941615835123165894, −0.837274506236724867628992284013, 1.68170968241167048561997460507, 3.08826808397347559353141315220, 4.28396576840732247907376857798, 5.35422352333761115161997447351, 6.42998185666481480970249950282, 7.15630673634734426369702843574, 7.961317285445614485398307614595, 8.807264756170572093439338571222, 10.65878827180455173802615877392, 11.11692663224714265961144666366

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