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Properties

Label 2-546-91.74-c1-0-16
Degree $2$
Conductor $546$
Sign $-0.0580 + 0.998i$
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 + (−0.5 − 0.866i)3-s + 4-s + (−0.441 − 0.764i)5-s + (−0.5 − 0.866i)6-s + (0.369 − 2.61i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.441 − 0.764i)10-s + (−0.775 − 1.34i)11-s + (−0.5 − 0.866i)12-s + (2.13 − 2.90i)13-s + (0.369 − 2.61i)14-s + (−0.441 + 0.764i)15-s + 16-s − 7.17·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.197 − 0.341i)5-s + (−0.204 − 0.353i)6-s + (0.139 − 0.990i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.139 − 0.241i)10-s + (−0.233 − 0.405i)11-s + (−0.144 − 0.249i)12-s + (0.591 − 0.805i)13-s + (0.0988 − 0.700i)14-s + (−0.113 + 0.197i)15-s + 0.250·16-s − 1.74·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0580 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0580 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0580 + 0.998i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0580 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24627 - 1.32088i\)
\(L(\frac12)\) \(\approx\) \(1.24627 - 1.32088i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.369 + 2.61i)T \)
13 \( 1 + (-2.13 + 2.90i)T \)
good5 \( 1 + (0.441 + 0.764i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.775 + 1.34i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 7.17T + 17T^{2} \)
19 \( 1 + (2.37 - 4.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.29T + 23T^{2} \)
29 \( 1 + (-3.87 + 6.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.330T + 37T^{2} \)
41 \( 1 + (3.02 - 5.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.35 - 5.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.976 - 1.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.74 - 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.27T + 59T^{2} \)
61 \( 1 + (-7.17 + 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.75 - 6.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.00 - 8.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.93 - 3.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + (-5.79 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.96866502458667119315711081590, −9.954597053034263186958299386865, −8.429769433594877043548579017820, −7.896102210224936528321946495970, −6.69200959027473877299692886377, −6.08388981409262132811927742313, −4.78792031049237203251871724987, −4.02767022098173651880493993052, −2.59630183762964731391340034771, −0.892733128753621187788149517683, 2.15184368730451333108424620849, 3.30973445519537675043719688581, 4.61785409446431762100936562007, 5.17246549299940839618912358782, 6.55272851635679152454052019227, 6.94558969087238909543736203307, 8.732645374438572317701982487113, 9.011691957154237023095661708918, 10.53029975130224415779456002874, 11.10746290326000497339070858240

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