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Properties

Label 2-546-91.54-c1-0-8
Degree $2$
Conductor $546$
Sign $0.968 - 0.248i$
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  + (−0.707 + 0.707i)2-s + (0.866 + 0.5i)3-s − 1.00i·4-s + (1.41 − 0.379i)5-s + (−0.965 + 0.258i)6-s + (0.378 − 2.61i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.733 + 1.27i)10-s + (−3.64 + 0.977i)11-s + (0.500 − 0.866i)12-s + (3.50 + 0.851i)13-s + (1.58 + 2.11i)14-s + (1.41 + 0.379i)15-s − 1.00·16-s + 7.07·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.499 + 0.288i)3-s − 0.500i·4-s + (0.633 − 0.169i)5-s + (−0.394 + 0.105i)6-s + (0.143 − 0.989i)7-s + (0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (−0.231 + 0.401i)10-s + (−1.10 + 0.294i)11-s + (0.144 − 0.250i)12-s + (0.971 + 0.236i)13-s + (0.423 + 0.566i)14-s + (0.365 + 0.0980i)15-s − 0.250·16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51101 + 0.190660i\)
\(L(\frac12)\) \(\approx\) \(1.51101 + 0.190660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.378 + 2.61i)T \)
13 \( 1 + (-3.50 - 0.851i)T \)
good5 \( 1 + (-1.41 + 0.379i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.64 - 0.977i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 + (-1.26 + 4.70i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 2.19iT - 23T^{2} \)
29 \( 1 + (-1.51 - 2.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.35 + 8.79i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.06 - 3.06i)T + 37iT^{2} \)
41 \( 1 + (0.520 - 1.94i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.52 - 2.03i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.85 - 10.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.155 + 0.268i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.86 - 1.86i)T - 59iT^{2} \)
61 \( 1 + (1.74 - 1.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.70 + 10.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.304 + 1.13i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.8 + 3.17i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.15 - 14.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \)
89 \( 1 + (-1.60 + 1.60i)T - 89iT^{2} \)
97 \( 1 + (3.84 - 1.03i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.52087755831545903105449938030, −9.863068389848898199023710593274, −9.177725463687302690847091296341, −7.959188216059178792621582617084, −7.59843790841210058844664628338, −6.32430531712310529647553962194, −5.32496296204707013850460404054, −4.27609518534525500302551244785, −2.83994597313608631388523218834, −1.20861013534583512397960691856, 1.47469354849919936538850742238, 2.66499557861599570857960362022, 3.54114836129561638196798179549, 5.42326039022552903415614678545, 6.02860172739838524659121963941, 7.55918783502599987957824590764, 8.225233409999222641266231770080, 8.947989384899241630400112659475, 10.04451913945717180382949500965, 10.42790021789206359471392643565

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