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Properties

Label 2-546-273.32-c1-0-36
Degree $2$
Conductor $546$
Sign $-0.924 + 0.380i$
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 + (−1.24 − 1.20i)3-s + 1.00i·4-s + (0.505 − 1.88i)5-s + (−0.0323 − 1.73i)6-s + (−2.61 + 0.408i)7-s + (−0.707 + 0.707i)8-s + (0.111 + 2.99i)9-s + (1.69 − 0.977i)10-s + (−3.42 − 0.918i)11-s + (1.20 − 1.24i)12-s + (−1.12 − 3.42i)13-s + (−2.13 − 1.55i)14-s + (−2.89 + 1.74i)15-s − 1.00·16-s − 2.53·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.720 − 0.693i)3-s + 0.500i·4-s + (0.226 − 0.844i)5-s + (−0.0131 − 0.706i)6-s + (−0.988 + 0.154i)7-s + (−0.250 + 0.250i)8-s + (0.0372 + 0.999i)9-s + (0.535 − 0.309i)10-s + (−1.03 − 0.276i)11-s + (0.346 − 0.360i)12-s + (−0.313 − 0.949i)13-s + (−0.571 − 0.416i)14-s + (−0.748 + 0.451i)15-s − 0.250·16-s − 0.615·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.924 + 0.380i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.924 + 0.380i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0606458 - 0.307101i\)
\(L(\frac12)\) \(\approx\) \(0.0606458 - 0.307101i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.24 + 1.20i)T \)
7 \( 1 + (2.61 - 0.408i)T \)
13 \( 1 + (1.12 + 3.42i)T \)
good5 \( 1 + (-0.505 + 1.88i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.42 + 0.918i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 2.53T + 17T^{2} \)
19 \( 1 + (-1.58 - 5.91i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.64T + 23T^{2} \)
29 \( 1 + (7.81 + 4.51i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.215 + 0.803i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.90 + 3.90i)T + 37iT^{2} \)
41 \( 1 + (-6.19 + 1.66i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.42 - 2.55i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (11.1 + 2.98i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.87 - 2.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.16 + 7.16i)T - 59iT^{2} \)
61 \( 1 + (-5.60 + 9.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.5 - 3.08i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.52 + 5.69i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.3 - 2.77i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.90 - 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.859 - 0.859i)T + 83iT^{2} \)
89 \( 1 + (-1.99 + 1.99i)T - 89iT^{2} \)
97 \( 1 + (-6.95 - 1.86i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.41579862564586518070511367223, −9.598585427969081311203311442417, −8.235918151103550123865209599726, −7.72911590870762273073808755700, −6.54864857664525525601343140460, −5.62763040748567294584381909545, −5.27768012878304708568444588103, −3.74840653559916977329368009010, −2.22890344259038484055876710281, −0.15296664764275581103913171480, 2.41502877467492596984379231279, 3.47232646025510131226302697393, 4.54738019367028837320513262426, 5.52416440050445637477056669995, 6.57249859630660882102531134511, 7.10853802076252557466741533439, 8.978663659098199129328440740707, 9.830654133298438678192906826115, 10.34670457217971257230182639496, 11.15249308553276744773654035629

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