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Properties

Label 2-546-273.242-c1-0-7
Degree $2$
Conductor $546$
Sign $0.422 - 0.906i$
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  + (−0.965 − 0.258i)2-s + (1.69 + 0.359i)3-s + (0.866 + 0.499i)4-s + (−3.75 − 1.00i)5-s + (−1.54 − 0.785i)6-s + (1.18 + 2.36i)7-s + (−0.707 − 0.707i)8-s + (2.74 + 1.21i)9-s + (3.36 + 1.94i)10-s + (1.60 − 0.429i)11-s + (1.28 + 1.15i)12-s + (−2.02 + 2.98i)13-s + (−0.529 − 2.59i)14-s + (−5.99 − 3.05i)15-s + (0.500 + 0.866i)16-s + (−0.0575 + 0.0997i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.978 + 0.207i)3-s + (0.433 + 0.249i)4-s + (−1.67 − 0.449i)5-s + (−0.630 − 0.320i)6-s + (0.446 + 0.894i)7-s + (−0.249 − 0.249i)8-s + (0.913 + 0.405i)9-s + (1.06 + 0.614i)10-s + (0.483 − 0.129i)11-s + (0.371 + 0.334i)12-s + (−0.560 + 0.828i)13-s + (−0.141 − 0.692i)14-s + (−1.54 − 0.787i)15-s + (0.125 + 0.216i)16-s + (−0.0139 + 0.0241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.422 - 0.906i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.422 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899189 + 0.573016i\)
\(L(\frac12)\) \(\approx\) \(0.899189 + 0.573016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-1.69 - 0.359i)T \)
7 \( 1 + (-1.18 - 2.36i)T \)
13 \( 1 + (2.02 - 2.98i)T \)
good5 \( 1 + (3.75 + 1.00i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.60 + 0.429i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.0575 - 0.0997i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.398 - 1.48i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.63 - 6.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.96iT - 29T^{2} \)
31 \( 1 + (0.532 - 0.142i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.30 - 0.885i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.86 - 7.86i)T - 41iT^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 + (-0.604 + 2.25i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.26 - 1.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.02 + 1.34i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.22 - 3.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.47 + 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.46 - 7.46i)T - 71iT^{2} \)
73 \( 1 + (1.57 + 5.85i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.63 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.41 + 3.41i)T - 83iT^{2} \)
89 \( 1 + (-0.0126 + 0.0472i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.27 + 4.27i)T + 97iT^{2} \)
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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

−11.08570989780901762842920421423, −9.808579339268584108569744889696, −8.856175303059215825568362524837, −8.578590077464397396407343084605, −7.62556066981640647849428486534, −6.98133096277302854387878376302, −5.08892514985047574122173183101, −4.04416068777127940205153140097, −3.11618805479052904643619462132, −1.62329552049453234776584103770, 0.73625916035460885299222218116, 2.66126581751852586716844505628, 3.77324280887123247480866423057, 4.64459136975930880925297144600, 6.75376056877397386601000665710, 7.25686559411191018305033799134, 8.042730152962628152850555017601, 8.486180839083100499711049981370, 9.720794442811671686414686459337, 10.59507873584841343861174994596

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