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Properties

Label 2-546-39.32-c1-0-25
Degree $2$
Conductor $546$
Sign $-0.984 - 0.173i$
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 + (0.504 − 1.65i)3-s + (0.866 − 0.499i)4-s + (−2.02 − 2.02i)5-s + (−0.0581 + 1.73i)6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + (−2.49 − 1.67i)9-s + (2.47 + 1.42i)10-s + (−0.522 − 1.95i)11-s + (−0.391 − 1.68i)12-s + (−2.79 + 2.27i)13-s + i·14-s + (−4.37 + 2.33i)15-s + (0.500 − 0.866i)16-s + (0.857 + 1.48i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.291 − 0.956i)3-s + (0.433 − 0.249i)4-s + (−0.904 − 0.904i)5-s + (−0.0237 + 0.706i)6-s + (0.0978 − 0.365i)7-s + (−0.249 + 0.249i)8-s + (−0.830 − 0.557i)9-s + (0.783 + 0.452i)10-s + (−0.157 − 0.588i)11-s + (−0.113 − 0.487i)12-s + (−0.775 + 0.631i)13-s + 0.267i·14-s + (−1.12 + 0.601i)15-s + (0.125 − 0.216i)16-s + (0.207 + 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0415348 + 0.476523i\)
\(L(\frac12)\) \(\approx\) \(0.0415348 + 0.476523i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.504 + 1.65i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (2.79 - 2.27i)T \)
good5 \( 1 + (2.02 + 2.02i)T + 5iT^{2} \)
11 \( 1 + (0.522 + 1.95i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.857 - 1.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.291 - 0.0781i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.189 + 0.328i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.49 - 0.862i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.01 - 2.01i)T - 31iT^{2} \)
37 \( 1 + (8.36 - 2.24i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.95 - 0.524i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.58 + 3.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.62 + 5.62i)T - 47iT^{2} \)
53 \( 1 - 5.57iT - 53T^{2} \)
59 \( 1 + (10.1 + 2.72i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.94 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.287 - 1.07i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.436 - 1.62i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (8.83 + 8.83i)T + 73iT^{2} \)
79 \( 1 + 3.44T + 79T^{2} \)
83 \( 1 + (11.1 + 11.1i)T + 83iT^{2} \)
89 \( 1 + (4.32 + 16.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.6 - 3.12i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.29877767019404551041130531560, −8.996017677705083382497375076945, −8.576505608097508995403605588975, −7.65978492366539055974545262420, −7.10951355748106562804747024061, −5.93499501309526270928564709903, −4.68186624789392272788973964412, −3.29559109177316103416342963561, −1.69057126715403170735452810734, −0.32393979797322983752023336934, 2.49661872106075915032649613932, 3.32127718276026811586928978144, 4.49065538687987908389841980851, 5.68375314027309180317640678682, 7.19210181890764162059100459210, 7.72517092157642237189473868970, 8.715853061744476356873166851642, 9.631044292206745834789083691316, 10.35362434317270272196753779514, 11.02450688340396842903234631094

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