Properties

Label 2-546-273.254-c1-0-19
Degree $2$
Conductor $546$
Sign $0.452 - 0.891i$
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 + (1.72 − 0.117i)3-s + (0.866 + 0.499i)4-s + (1.12 + 4.19i)5-s + (1.69 + 0.333i)6-s + (−2.63 + 0.258i)7-s + (0.707 + 0.707i)8-s + (2.97 − 0.406i)9-s + 4.34i·10-s + (−1.35 − 1.35i)11-s + (1.55 + 0.762i)12-s + (−0.217 − 3.59i)13-s + (−2.61 − 0.431i)14-s + (2.43 + 7.11i)15-s + (0.500 + 0.866i)16-s + (0.351 − 0.609i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.997 − 0.0678i)3-s + (0.433 + 0.249i)4-s + (0.502 + 1.87i)5-s + (0.693 + 0.136i)6-s + (−0.995 + 0.0978i)7-s + (0.249 + 0.249i)8-s + (0.990 − 0.135i)9-s + 1.37i·10-s + (−0.407 − 0.407i)11-s + (0.448 + 0.220i)12-s + (−0.0602 − 0.998i)13-s + (−0.697 − 0.115i)14-s + (0.629 + 1.83i)15-s + (0.125 + 0.216i)16-s + (0.0852 − 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46622 + 1.51332i\)
\(L(\frac12)\) \(\approx\) \(2.46622 + 1.51332i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.117i)T \)
7 \( 1 + (2.63 - 0.258i)T \)
13 \( 1 + (0.217 + 3.59i)T \)
good5 \( 1 + (-1.12 - 4.19i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.35 + 1.35i)T + 11iT^{2} \)
17 \( 1 + (-0.351 + 0.609i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.51 + 2.51i)T + 19iT^{2} \)
23 \( 1 + (-0.914 - 1.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.59 - 3.80i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.94 + 7.26i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.43 - 0.383i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.735 + 2.74i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.30 + 1.90i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.438 - 0.117i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.96 + 4.02i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.58 - 1.76i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + (-1.89 - 1.89i)T + 67iT^{2} \)
71 \( 1 + (12.7 + 3.41i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (13.7 + 3.67i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.155 - 0.269i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.970 + 0.970i)T - 83iT^{2} \)
89 \( 1 + (3.19 - 11.9i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.18 - 8.14i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.64226117218081121069290047812, −10.29164493498354486049661982298, −9.310994516341802005159622104034, −8.042515992356184397946911016559, −7.14104693314607961462185322530, −6.53375137947597737332454107206, −5.61920116165866864195662767103, −3.86740270858376381085664093828, −2.89906164484799988682369387843, −2.58463388785983823111667727440, 1.43740717525642342997953559709, 2.64722675995761018921599512695, 4.13957514581918763249147604394, 4.65037360706947493354144354885, 5.89231854017960399263042196448, 6.93861650711239672636657855086, 8.240170719050159828368916668256, 8.908060644426246437107466661481, 9.742425599393255833027318704223, 10.29407227413813441832477236585

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