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Properties

Label 2-546-273.62-c1-0-20
Degree $2$
Conductor $546$
Sign $0.382 + 0.923i$
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 + (0.334 − 1.69i)3-s + (−0.499 + 0.866i)4-s + 2.94i·5-s + (−1.63 + 0.559i)6-s + (1.55 − 2.14i)7-s + 0.999·8-s + (−2.77 − 1.13i)9-s + (2.55 − 1.47i)10-s + (1.48 + 2.57i)11-s + (1.30 + 1.13i)12-s + (2.33 + 2.75i)13-s + (−2.63 − 0.272i)14-s + (5.01 + 0.987i)15-s + (−0.5 − 0.866i)16-s + (2.27 − 3.93i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.193 − 0.981i)3-s + (−0.249 + 0.433i)4-s + 1.31i·5-s + (−0.669 + 0.228i)6-s + (0.586 − 0.809i)7-s + 0.353·8-s + (−0.925 − 0.379i)9-s + (0.807 − 0.466i)10-s + (0.448 + 0.776i)11-s + (0.376 + 0.329i)12-s + (0.646 + 0.762i)13-s + (−0.703 − 0.0727i)14-s + (1.29 + 0.254i)15-s + (−0.125 − 0.216i)16-s + (0.551 − 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15533 - 0.772024i\)
\(L(\frac12)\) \(\approx\) \(1.15533 - 0.772024i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.334 + 1.69i)T \)
7 \( 1 + (-1.55 + 2.14i)T \)
13 \( 1 + (-2.33 - 2.75i)T \)
good5 \( 1 - 2.94iT - 5T^{2} \)
11 \( 1 + (-1.48 - 2.57i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.27 + 3.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.30 + 5.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.82 + 1.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.262 - 0.151i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.31T + 31T^{2} \)
37 \( 1 + (5.19 - 2.99i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.22 + 2.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.53 - 7.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.77iT - 47T^{2} \)
53 \( 1 + 1.95iT - 53T^{2} \)
59 \( 1 + (-5.73 - 3.30i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.23 + 3.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.91 - 3.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.69 + 8.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.10T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 5.68iT - 83T^{2} \)
89 \( 1 + (12.8 - 7.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.31 + 9.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.80864981886754841183937486564, −9.849115350650344275957679074162, −8.914495747286620939399081456696, −7.78474684125600066152640127139, −7.03611445681139948565984952388, −6.62094459849352660304457515153, −4.82839041586608356882419247986, −3.44963733214501706791500455840, −2.48606095638433838855287308651, −1.16751213215515251535490521341, 1.30915355192215055591510821296, 3.39344748491396048653998866438, 4.54061418463781069772666018872, 5.60963713747436459363997401822, 5.85115976125932781132088086506, 7.942698932061242568406734350042, 8.429148338218327234473388067667, 8.951274857489568205231180862741, 9.866645526654265009024898834968, 10.75759886539582074551006564977

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