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Properties

Label 2-546-91.54-c1-0-15
Degree $2$
Conductor $546$
Sign $0.707 + 0.707i$
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 + (1.56 − 0.420i)5-s + (0.965 − 0.258i)6-s + (2.44 − 1.01i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.811 − 1.40i)10-s + (−0.797 + 0.213i)11-s + (0.500 − 0.866i)12-s + (−3.50 − 0.855i)13-s + (1.01 − 2.44i)14-s + (1.56 + 0.420i)15-s − 1.00·16-s + 4.29·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.499 + 0.288i)3-s − 0.500i·4-s + (0.701 − 0.187i)5-s + (0.394 − 0.105i)6-s + (0.924 − 0.382i)7-s + (−0.250 − 0.250i)8-s + (0.166 + 0.288i)9-s + (0.256 − 0.444i)10-s + (−0.240 + 0.0643i)11-s + (0.144 − 0.250i)12-s + (−0.971 − 0.237i)13-s + (0.270 − 0.653i)14-s + (0.404 + 0.108i)15-s − 0.250·16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.707 + 0.707i$
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: \((2,\ 546,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35852 - 0.976941i\)
\(L(\frac12)\) \(\approx\) \(2.35852 - 0.976941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + 1.01i)T \)
13 \( 1 + (3.50 + 0.855i)T \)
good5 \( 1 + (-1.56 + 0.420i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.797 - 0.213i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 4.29T + 17T^{2} \)
19 \( 1 + (0.170 - 0.637i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.381iT - 23T^{2} \)
29 \( 1 + (0.856 + 1.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.08 - 4.05i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.265 - 0.265i)T + 37iT^{2} \)
41 \( 1 + (-0.129 + 0.484i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.60 - 2.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.54 + 13.2i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.29 - 3.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.09 - 1.09i)T - 59iT^{2} \)
61 \( 1 + (10.6 - 6.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.94 - 7.24i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.09 - 4.09i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.0 + 2.97i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.74 - 6.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.17 - 5.17i)T + 83iT^{2} \)
89 \( 1 + (-7.47 + 7.47i)T - 89iT^{2} \)
97 \( 1 + (15.6 - 4.19i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.49477105674000478242937200474, −10.03175903279529169785508267409, −9.135997229831755127312740197289, −8.030277718102304817616585128137, −7.20237138843124902361576952903, −5.65887325151873498470735831624, −5.02271709379201873286845801425, −3.95367552145806505943870399401, −2.67424963419861397483504028198, −1.53178841153744960943407090724, 1.90060869222594525530534270213, 2.94370455860068471899653105245, 4.43034560113041366129726837963, 5.40498712966307597200343084046, 6.24136239001865539673394003972, 7.47698349886035951483642646251, 7.950580039076195486514522760501, 9.084203791491231583801948505235, 9.851698887716269280728122319920, 10.99181750469163824415118434736

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