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Properties

Label 2-546-91.74-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.0598 + 0.998i$
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 + (−1.97 − 3.42i)5-s + (0.5 + 0.866i)6-s + (−1.15 − 2.38i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−1.97 − 3.42i)10-s + (−2.45 − 4.25i)11-s + (0.5 + 0.866i)12-s + (−3.39 + 1.21i)13-s + (−1.15 − 2.38i)14-s + (1.97 − 3.42i)15-s + 16-s + 0.140·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.883 − 1.52i)5-s + (0.204 + 0.353i)6-s + (−0.435 − 0.900i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.624 − 1.08i)10-s + (−0.740 − 1.28i)11-s + (0.144 + 0.249i)12-s + (−0.941 + 0.338i)13-s + (−0.307 − 0.636i)14-s + (0.509 − 0.883i)15-s + 0.250·16-s + 0.0340·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0598 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0598 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0598 + 0.998i$
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: \((2,\ 546,\ (\ :1/2),\ -0.0598 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12960 - 1.19933i\)
\(L(\frac12)\) \(\approx\) \(1.12960 - 1.19933i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 + 2.38i)T \)
13 \( 1 + (3.39 - 1.21i)T \)
good5 \( 1 + (1.97 + 3.42i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.45 + 4.25i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.140T + 17T^{2} \)
19 \( 1 + (-0.388 + 0.672i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 9.53T + 23T^{2} \)
29 \( 1 + (0.629 - 1.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.67 - 2.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + (-4.65 + 8.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.541 + 0.937i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.33 - 5.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.53 + 9.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.431T + 59T^{2} \)
61 \( 1 + (-4.14 + 7.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.09 + 7.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.93 - 3.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.0817 - 0.141i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.17 - 3.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + 1.07T + 89T^{2} \)
97 \( 1 + (6.54 + 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84946239615057978145936274751, −9.593423035510658132776804774669, −8.793751092197079437966045578620, −7.88833813144627472330140980324, −7.07648327128693425771023056594, −5.50669382646861067347393800206, −4.79398177847373631261426592140, −3.96952982485927793554815017021, −2.97650281298064673523593771498, −0.72278464043229839220151938478, 2.62906258487138382258670146514, 2.78642900368569589272471506941, 4.27595616495077692720184657183, 5.52564252940819541935750707124, 6.67340963511480690161624767698, 7.32514335992778650553085188031, 7.889272408967896093407550897771, 9.401658406215320030649319876061, 10.30455079487262753251669817984, 11.24270379299287940950150731648

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