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Properties

Label 2-546-273.17-c1-0-35
Degree $2$
Conductor $546$
Sign $0.811 + 0.583i$
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 + (1.67 − 0.438i)3-s + 4-s + (1.57 − 0.908i)5-s + (1.67 − 0.438i)6-s + (−2.47 − 0.947i)7-s + 8-s + (2.61 − 1.46i)9-s + (1.57 − 0.908i)10-s + (−1.02 − 1.77i)11-s + (1.67 − 0.438i)12-s + (−3.57 − 0.463i)13-s + (−2.47 − 0.947i)14-s + (2.23 − 2.21i)15-s + 16-s + 5.68·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.967 − 0.253i)3-s + 0.5·4-s + (0.703 − 0.406i)5-s + (0.684 − 0.178i)6-s + (−0.933 − 0.358i)7-s + 0.353·8-s + (0.871 − 0.489i)9-s + (0.497 − 0.287i)10-s + (−0.309 − 0.535i)11-s + (0.483 − 0.126i)12-s + (−0.991 − 0.128i)13-s + (−0.660 − 0.253i)14-s + (0.577 − 0.571i)15-s + 0.250·16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.811 + 0.583i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.811 + 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.86370 - 0.922925i\)
\(L(\frac12)\) \(\approx\) \(2.86370 - 0.922925i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 + 0.438i)T \)
7 \( 1 + (2.47 + 0.947i)T \)
13 \( 1 + (3.57 + 0.463i)T \)
good5 \( 1 + (-1.57 + 0.908i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.02 + 1.77i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.68T + 17T^{2} \)
19 \( 1 + (0.796 - 1.38i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.73iT - 23T^{2} \)
29 \( 1 + (-0.724 - 0.418i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.97 - 6.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.21iT - 37T^{2} \)
41 \( 1 + (-0.397 - 0.229i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.836 - 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.94 - 1.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.497 - 0.286i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.19iT - 59T^{2} \)
61 \( 1 + (5.97 + 3.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.46 - 2.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.24 - 7.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.11 + 8.86i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.68 + 6.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 - 3.86iT - 89T^{2} \)
97 \( 1 + (1.72 + 2.98i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49654572263595776891888683970, −9.749343916904733617625346467594, −9.144957097188330059459338281814, −7.79450806011310378290939161997, −7.21000246895694164007656658819, −5.99341819457373175284789309354, −5.17102364853822235434930872969, −3.66360583726351617129748091458, −2.99684868385960683853939616134, −1.56171617527279828577172972417, 2.28754318413548731401292433700, 2.84277863937365266174416274102, 4.10281878231057157399301560852, 5.20888712817169432896026077394, 6.32505221607361622002961690608, 7.17308952441813084253076368122, 8.146095394970072091289476076087, 9.412285768018712912477546861214, 9.956365562511176923636352781853, 10.58211564303303064551092570515

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