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Properties

Label 2-546-21.5-c1-0-12
Degree $2$
Conductor $546$
Sign $0.0333 - 0.999i$
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  + (0.866 + 0.5i)2-s + (1.48 − 0.899i)3-s + (0.499 + 0.866i)4-s + (−1.85 + 3.21i)5-s + (1.73 − 0.0386i)6-s + (−1.15 + 2.37i)7-s + 0.999i·8-s + (1.38 − 2.66i)9-s + (−3.21 + 1.85i)10-s + (−1.05 + 0.610i)11-s + (1.51 + 0.832i)12-s − i·13-s + (−2.19 + 1.48i)14-s + (0.143 + 6.43i)15-s + (−0.5 + 0.866i)16-s + (2.08 + 3.60i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.854 − 0.519i)3-s + (0.249 + 0.433i)4-s + (−0.831 + 1.43i)5-s + (0.706 − 0.0157i)6-s + (−0.437 + 0.899i)7-s + 0.353i·8-s + (0.460 − 0.887i)9-s + (−1.01 + 0.587i)10-s + (−0.318 + 0.183i)11-s + (0.438 + 0.240i)12-s − 0.277i·13-s + (−0.585 + 0.396i)14-s + (0.0370 + 1.66i)15-s + (−0.125 + 0.216i)16-s + (0.504 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0333 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0333 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0333 - 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0333 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56763 + 1.51615i\)
\(L(\frac12)\) \(\approx\) \(1.56763 + 1.51615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.48 + 0.899i)T \)
7 \( 1 + (1.15 - 2.37i)T \)
13 \( 1 + iT \)
good5 \( 1 + (1.85 - 3.21i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.05 - 0.610i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.08 - 3.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.91 - 2.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.756 + 0.436i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.39iT - 29T^{2} \)
31 \( 1 + (2.49 - 1.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.546 - 0.946i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + (-4.05 + 7.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.98 - 4.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.97 + 5.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.35 - 3.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.28 + 12.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.92iT - 71T^{2} \)
73 \( 1 + (1.89 - 1.09i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.29 + 2.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.05T + 83T^{2} \)
89 \( 1 + (-1.01 + 1.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−11.15754478241175498927466224477, −10.16581045799535646418425348506, −9.105183664977278952141029219366, −7.82951908599790652972656409664, −7.62664612441587359115118697022, −6.49422138768056640038712625379, −5.77650740102554239555645515058, −3.97658529415744541939892500754, −3.19242691630711002150181235136, −2.38625589860878169897898514003, 1.00563737081225922571910593400, 2.91090570573380576673970040350, 3.95060193351809564887545009279, 4.58641759716085415602305819005, 5.50384392127958494606572082722, 7.31707799298479760928781026464, 7.81438887121119564964125973375, 9.119122907517037351138182654267, 9.478450606368197733964027450378, 10.66741750431418735976412435070

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