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Properties

Label 2-546-91.74-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.100 - 0.994i$
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 + (1.10 + 1.91i)5-s + (−0.5 − 0.866i)6-s + (1.19 + 2.36i)7-s − 8-s + (−0.499 + 0.866i)9-s + (−1.10 − 1.91i)10-s + (0.527 + 0.914i)11-s + (0.5 + 0.866i)12-s + (3.18 − 1.69i)13-s + (−1.19 − 2.36i)14-s + (−1.10 + 1.91i)15-s + 16-s − 0.944·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (0.493 + 0.854i)5-s + (−0.204 − 0.353i)6-s + (0.450 + 0.892i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.348 − 0.604i)10-s + (0.159 + 0.275i)11-s + (0.144 + 0.249i)12-s + (0.883 − 0.469i)13-s + (−0.318 − 0.631i)14-s + (−0.284 + 0.493i)15-s + 0.250·16-s − 0.229·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.100 - 0.994i$
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: \((2,\ 546,\ (\ :1/2),\ -0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859154 + 0.950486i\)
\(L(\frac12)\) \(\approx\) \(0.859154 + 0.950486i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.19 - 2.36i)T \)
13 \( 1 + (-3.18 + 1.69i)T \)
good5 \( 1 + (-1.10 - 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.527 - 0.914i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.944T + 17T^{2} \)
19 \( 1 + (1.96 - 3.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.22T + 23T^{2} \)
29 \( 1 + (-0.888 + 1.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.63 + 6.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.26T + 37T^{2} \)
41 \( 1 + (1.63 - 2.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.537 - 0.930i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.42 - 4.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.94 - 8.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 + (0.0382 - 0.0661i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.85 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.20 + 7.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.57 + 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.00 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.77T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (8.99 + 15.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.80831439630462588415873561543, −10.10170074217295868586042450249, −9.375804271978717223484712743076, −8.380809189589915509642299115636, −7.80385830739139410522240048496, −6.31615041939090113504711161063, −5.85484427282414761408772174007, −4.28053468509404604656524056547, −2.90965862046940005784072720396, −1.93479114242809629275524679903, 0.955500380296911027281977551228, 1.99307053318053842454897535633, 3.71278363059800471022160630826, 4.95430230112104486887056632190, 6.26373639232111177284025149032, 7.01591905803119439710572821096, 8.177932295991268341749220758026, 8.660352986020552054441330064070, 9.518840250655051980417888039545, 10.52435103253008227033677382420

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