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Properties

Label 2-546-91.24-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.864 + 0.501i$
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 − i·3-s + (0.866 − 0.499i)4-s + (−2.85 − 0.765i)5-s + (−0.258 − 0.965i)6-s + (0.565 − 2.58i)7-s + (0.707 − 0.707i)8-s − 9-s − 2.95·10-s + (−3.08 + 3.08i)11-s + (−0.499 − 0.866i)12-s + (−1.11 − 3.42i)13-s + (−0.123 − 2.64i)14-s + (−0.765 + 2.85i)15-s + (0.500 − 0.866i)16-s + (−0.464 − 0.804i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s − 0.577i·3-s + (0.433 − 0.249i)4-s + (−1.27 − 0.342i)5-s + (−0.105 − 0.394i)6-s + (0.213 − 0.976i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s − 0.935·10-s + (−0.931 + 0.931i)11-s + (−0.144 − 0.249i)12-s + (−0.308 − 0.951i)13-s + (−0.0329 − 0.706i)14-s + (−0.197 + 0.737i)15-s + (0.125 − 0.216i)16-s + (−0.112 − 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.864 + 0.501i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.864 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328638 - 1.22105i\)
\(L(\frac12)\) \(\approx\) \(0.328638 - 1.22105i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (-0.565 + 2.58i)T \)
13 \( 1 + (1.11 + 3.42i)T \)
good5 \( 1 + (2.85 + 0.765i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.08 - 3.08i)T - 11iT^{2} \)
17 \( 1 + (0.464 + 0.804i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.118 + 0.118i)T - 19iT^{2} \)
23 \( 1 + (1.39 + 0.806i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.94 - 3.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.23 + 8.33i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.80 + 6.72i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-9.71 - 2.60i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.51 - 2.60i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.34 + 8.74i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.13 + 1.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.903 + 3.37i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 9.81iT - 61T^{2} \)
67 \( 1 + (-3.86 - 3.86i)T + 67iT^{2} \)
71 \( 1 + (-7.17 + 1.92i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.21 + 1.66i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.30 - 7.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.81 + 1.81i)T - 83iT^{2} \)
89 \( 1 + (11.7 - 3.14i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.87 + 10.7i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.74427310356233036484359638371, −9.776530383064338206509578485605, −8.233307912126743613033039262301, −7.56917634543550188348963293534, −7.11001508287826569603993183623, −5.60077356564485688941584131558, −4.59634884453774878406789970663, −3.78784980449585668722534268907, −2.42342635284816682582885095697, −0.56815160663209430372745311569, 2.56582863722192543526585871037, 3.54340719544367729183016904339, 4.54111072247132570462799257365, 5.45838215739154316935480020005, 6.48102990469029285954033574152, 7.68429008639570010296870898111, 8.347631353373523233602060510591, 9.282710178609001055844370529289, 10.67173925947188299329392028071, 11.23717979676231418860985095725

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