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Properties

Label 2-546-91.9-c1-0-5
Degree $2$
Conductor $546$
Sign $-0.886 - 0.462i$
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  + (0.5 + 0.866i)2-s + 3-s + (−0.499 + 0.866i)4-s + (−1.15 + 1.99i)5-s + (0.5 + 0.866i)6-s + (−0.964 + 2.46i)7-s − 0.999·8-s + 9-s − 2.30·10-s − 5.85·11-s + (−0.499 + 0.866i)12-s + (3.58 − 0.349i)13-s + (−2.61 + 0.396i)14-s + (−1.15 + 1.99i)15-s + (−0.5 − 0.866i)16-s + (0.624 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 0.577·3-s + (−0.249 + 0.433i)4-s + (−0.514 + 0.891i)5-s + (0.204 + 0.353i)6-s + (−0.364 + 0.931i)7-s − 0.353·8-s + 0.333·9-s − 0.728·10-s − 1.76·11-s + (−0.144 + 0.249i)12-s + (0.995 − 0.0968i)13-s + (−0.699 + 0.105i)14-s + (−0.297 + 0.514i)15-s + (−0.125 − 0.216i)16-s + (0.151 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.886 - 0.462i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341435 + 1.39244i\)
\(L(\frac12)\) \(\approx\) \(0.341435 + 1.39244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 - T \)
7 \( 1 + (0.964 - 2.46i)T \)
13 \( 1 + (-3.58 + 0.349i)T \)
good5 \( 1 + (1.15 - 1.99i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 5.85T + 11T^{2} \)
17 \( 1 + (-0.624 + 1.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 + (-0.972 - 1.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.12 + 3.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.62 - 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.52 - 6.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.45 + 2.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.38 - 9.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.13 - 3.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.05 - 7.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.58 - 6.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.71T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + (4.70 + 8.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.25 + 7.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.43 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 + (2.28 + 3.95i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.71 + 2.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.09046640409947606079090780488, −10.35055636604987145572796118319, −9.203744608938757729337408885836, −8.233051136500178494725368097595, −7.73325240603277085743078030707, −6.61045428892891817385051176269, −5.79241910423582441121539094998, −4.60709768684538713574806093985, −3.23086291870881716671326850584, −2.62977286258131307789384477356, 0.68304513844620496861605701903, 2.40566558146356262592490175440, 3.70499929824039181065877845053, 4.42123281691841021408210298149, 5.50819217889464372244688632602, 6.85631198219381920389145820324, 8.090708489169244977352565999504, 8.497037599356341915678836307554, 9.698866508782207994620922878242, 10.56460002496935085472331771026

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