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Properties

Label 2-546-39.32-c1-0-21
Degree $2$
Conductor $546$
Sign $0.807 + 0.589i$
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 + (1.18 − 1.26i)3-s + (0.866 − 0.499i)4-s + (2.02 + 2.02i)5-s + (0.815 − 1.52i)6-s + (0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (−0.201 − 2.99i)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 + (4.95 − 0.166i)15-s + (0.500 − 0.866i)16-s + (−0.857 − 1.48i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.682 − 0.730i)3-s + (0.433 − 0.249i)4-s + (0.904 + 0.904i)5-s + (0.332 − 0.623i)6-s + (0.0978 − 0.365i)7-s + (0.249 − 0.249i)8-s + (−0.0671 − 0.997i)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 + (1.27 − 0.0429i)15-s + (0.125 − 0.216i)16-s + (−0.207 − 0.360i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.807 + 0.589i$
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: \((2,\ 546,\ (\ :1/2),\ 0.807 + 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.81983 - 0.919282i\)
\(L(\frac12)\) \(\approx\) \(2.81983 - 0.919282i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 1.26i)T \)
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

−10.72683953274647734783760798263, −9.845694062076639027673562522754, −9.146968076348625061054632892249, −7.70136197781379229457410007066, −6.91883801264105813819563323874, −6.39545884613806955032310489772, −5.08514515217382532153355792090, −3.75908980567086804931665869214, −2.59309919073887222933089368857, −1.79784707287868689428583775857, 1.94866041483134859142476294143, 3.11298964040180912628006584609, 4.34745462917796631026605113438, 5.29652556938896030639263100727, 5.83902610636528522642090990611, 7.35444368071776795884005739471, 8.438482231761856065018316472431, 9.050515291389033003876980204548, 9.934893576061187352072059772629, 10.77564462135340841328753206313

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