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Properties

Label 2-546-273.62-c1-0-7
Degree $2$
Conductor $546$
Sign $-0.522 - 0.852i$
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  + (0.5 + 0.866i)2-s + (0.702 − 1.58i)3-s + (−0.499 + 0.866i)4-s + 3.28i·5-s + (1.72 − 0.182i)6-s + (−2.38 + 1.14i)7-s − 0.999·8-s + (−2.01 − 2.22i)9-s + (−2.84 + 1.64i)10-s + (2.08 + 3.61i)11-s + (1.01 + 1.40i)12-s + (−3.60 + 0.0395i)13-s + (−2.18 − 1.49i)14-s + (5.19 + 2.30i)15-s + (−0.5 − 0.866i)16-s + (−3.84 + 6.66i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.405 − 0.913i)3-s + (−0.249 + 0.433i)4-s + 1.46i·5-s + (0.703 − 0.0745i)6-s + (−0.901 + 0.431i)7-s − 0.353·8-s + (−0.670 − 0.741i)9-s + (−0.899 + 0.519i)10-s + (0.628 + 1.08i)11-s + (0.294 + 0.404i)12-s + (−0.999 + 0.0109i)13-s + (−0.583 − 0.399i)14-s + (1.34 + 0.595i)15-s + (−0.125 − 0.216i)16-s + (−0.933 + 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693670 + 1.23940i\)
\(L(\frac12)\) \(\approx\) \(0.693670 + 1.23940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.702 + 1.58i)T \)
7 \( 1 + (2.38 - 1.14i)T \)
13 \( 1 + (3.60 - 0.0395i)T \)
good5 \( 1 - 3.28iT - 5T^{2} \)
11 \( 1 + (-2.08 - 3.61i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.84 - 6.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.26 + 2.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.82 + 2.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.55 + 2.05i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.97T + 31T^{2} \)
37 \( 1 + (-5.29 + 3.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.73 - 1.00i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.64 - 8.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 - 4.48iT - 53T^{2} \)
59 \( 1 + (1.69 + 0.978i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.38 - 3.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.107 - 0.0620i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.79 + 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 - 2.20T + 79T^{2} \)
83 \( 1 - 5.80iT - 83T^{2} \)
89 \( 1 + (-5.51 + 3.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.26 + 5.65i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.25591190629478266058824669729, −10.05633477494800171672465600943, −9.275009928160383119539001355809, −8.169726587187576945448509590331, −7.15882949895440009891022147424, −6.61698456341654309837847148470, −6.16767002083609367969633508311, −4.41982005872850772707745017320, −3.08546021512339399299522412565, −2.33401227175771380999276055014, 0.68570768071125638941961417798, 2.72350871149566642520072962441, 3.73861742570715561657784117947, 4.75411448114576716881788781432, 5.33176303797094714782345579035, 6.74777018436918696237689490377, 8.240913209552462914690145034377, 9.109494955907268621621590424871, 9.520909158162441118979286133406, 10.34727213540359073653526151222

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