Properties

Label 2-546-39.32-c1-0-26
Degree $2$
Conductor $546$
Sign $-0.634 + 0.773i$
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 + (0.563 − 1.63i)3-s + (0.866 − 0.499i)4-s + (−1.58 − 1.58i)5-s + (0.120 − 1.72i)6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−2.36 − 1.84i)9-s + (−1.93 − 1.11i)10-s + (−1.28 − 4.80i)11-s + (−0.331 − 1.70i)12-s + (0.738 + 3.52i)13-s + i·14-s + (−3.48 + 1.70i)15-s + (0.500 − 0.866i)16-s + (−1.25 − 2.17i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.325 − 0.945i)3-s + (0.433 − 0.249i)4-s + (−0.707 − 0.707i)5-s + (0.0490 − 0.705i)6-s + (−0.0978 + 0.365i)7-s + (0.249 − 0.249i)8-s + (−0.788 − 0.614i)9-s + (−0.613 − 0.353i)10-s + (−0.388 − 1.44i)11-s + (−0.0956 − 0.490i)12-s + (0.204 + 0.978i)13-s + 0.267i·14-s + (−0.899 + 0.439i)15-s + (0.125 − 0.216i)16-s + (−0.303 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.634 + 0.773i$
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: \((2,\ 546,\ (\ :1/2),\ -0.634 + 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.810404 - 1.71303i\)
\(L(\frac12)\) \(\approx\) \(0.810404 - 1.71303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.563 + 1.63i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-0.738 - 3.52i)T \)
good5 \( 1 + (1.58 + 1.58i)T + 5iT^{2} \)
11 \( 1 + (1.28 + 4.80i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.25 + 2.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.54 - 0.412i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.566 + 0.981i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.50 - 3.18i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.36 + 1.36i)T - 31iT^{2} \)
37 \( 1 + (-1.84 + 0.494i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.36 + 0.902i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.80 + 3.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.51 - 8.51i)T - 47iT^{2} \)
53 \( 1 + 1.61iT - 53T^{2} \)
59 \( 1 + (-5.35 - 1.43i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.56 - 2.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.18 + 11.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.709 + 2.64i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.64 - 8.64i)T + 73iT^{2} \)
79 \( 1 - 17.5T + 79T^{2} \)
83 \( 1 + (-4.87 - 4.87i)T + 83iT^{2} \)
89 \( 1 + (1.28 + 4.78i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.8 + 4.25i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.96043868057972552026379110380, −9.340486389738065507720942862614, −8.564474966798669794590994197269, −7.86617870445332928149046026962, −6.71924900135781006296454976437, −5.91464555431988483489167743091, −4.78898404351233464659239423366, −3.56247604774189599369799508445, −2.50859717253581797554159921065, −0.866786708019047538895542473170, 2.55705849747117594586701563563, 3.53285533071032487527477372227, 4.39423427101245318152554380945, 5.28389857679752591942927688333, 6.57749081231754342296733870266, 7.58600501220476688520152298107, 8.187637523975347921683372605970, 9.600350453790178424596213891488, 10.37276067343232470827555001075, 10.99034296772687531048664300371

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