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Properties

Label 2-546-273.62-c1-0-23
Degree $2$
Conductor $546$
Sign $0.593 - 0.805i$
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 + (1.68 − 0.405i)3-s + (−0.499 + 0.866i)4-s − 0.465i·5-s + (1.19 + 1.25i)6-s + (1.06 + 2.42i)7-s − 0.999·8-s + (2.67 − 1.36i)9-s + (0.403 − 0.232i)10-s + (1.39 + 2.41i)11-s + (−0.490 + 1.66i)12-s + (0.799 − 3.51i)13-s + (−1.56 + 2.13i)14-s + (−0.189 − 0.784i)15-s + (−0.5 − 0.866i)16-s + (−0.508 + 0.880i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.972 − 0.234i)3-s + (−0.249 + 0.433i)4-s − 0.208i·5-s + (0.487 + 0.512i)6-s + (0.400 + 0.916i)7-s − 0.353·8-s + (0.890 − 0.455i)9-s + (0.127 − 0.0736i)10-s + (0.421 + 0.729i)11-s + (−0.141 + 0.479i)12-s + (0.221 − 0.975i)13-s + (−0.419 + 0.569i)14-s + (−0.0488 − 0.202i)15-s + (−0.125 − 0.216i)16-s + (−0.123 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.593 - 0.805i$
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: \((2,\ 546,\ (\ :1/2),\ 0.593 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18065 + 1.10215i\)
\(L(\frac12)\) \(\approx\) \(2.18065 + 1.10215i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.405i)T \)
7 \( 1 + (-1.06 - 2.42i)T \)
13 \( 1 + (-0.799 + 3.51i)T \)
good5 \( 1 + 0.465iT - 5T^{2} \)
11 \( 1 + (-1.39 - 2.41i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.508 - 0.880i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.817 + 1.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.14 - 4.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.01 - 2.31i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.94T + 31T^{2} \)
37 \( 1 + (-4.51 + 2.60i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.35 + 5.40i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.24 - 5.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.277iT - 47T^{2} \)
53 \( 1 + 8.99iT - 53T^{2} \)
59 \( 1 + (8.80 + 5.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.2 + 6.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.308 - 0.178i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.01 + 8.68i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.46T + 73T^{2} \)
79 \( 1 + 4.98T + 79T^{2} \)
83 \( 1 + 12.2iT - 83T^{2} \)
89 \( 1 + (10.1 - 5.85i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.82 - 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.97651708676261955708956026674, −9.608431861228921863588264860757, −9.081122071950496962706837765474, −8.081721899046625983686642589196, −7.57758468638805969800666116202, −6.38970149847857187119508188468, −5.39183297692046936549119534541, −4.28720109974333601361866310436, −3.14261901733812599097717414989, −1.85052201909700988991248250851, 1.44836139568584936632119306697, 2.77311494388112834876445394654, 3.98049197007479064120449103737, 4.43531504597479263294480166382, 6.05025662691940157896545450920, 7.16019331292599673705165386046, 8.140150940994415522973778198787, 9.026174537867886728893879288856, 9.853301100194972512529166747992, 10.71167559416107041350882173228

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