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Properties

Label 2-546-39.32-c1-0-6
Degree $2$
Conductor $546$
Sign $0.878 - 0.478i$
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 + (1.13 − 1.30i)3-s + (0.866 − 0.499i)4-s + (1.58 + 1.58i)5-s + (−0.759 + 1.55i)6-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (−0.414 − 2.97i)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 + (3.86 − 0.268i)15-s + (0.500 − 0.866i)16-s + (1.25 + 2.17i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.656 − 0.754i)3-s + (0.433 − 0.249i)4-s + (0.707 + 0.707i)5-s + (−0.310 + 0.635i)6-s + (−0.0978 + 0.365i)7-s + (−0.249 + 0.249i)8-s + (−0.138 − 0.990i)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 + (0.998 − 0.0694i)15-s + (0.125 − 0.216i)16-s + (0.303 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.878 - 0.478i$
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: \((2,\ 546,\ (\ :1/2),\ 0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43298 + 0.365071i\)
\(L(\frac12)\) \(\approx\) \(1.43298 + 0.365071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 + 1.30i)T \)
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

−10.66029121047852969736927512693, −9.579265123573298443683779914531, −9.347411249458228278591497891754, −8.155899522722147156843644903936, −7.22196254959617792452711290481, −6.63187026604951519598474482593, −5.75627150913422371175810987837, −3.95496628236803292679502798994, −2.42137509299210479585126942556, −1.72803405470343090200905408004, 1.10075437281015851907529910165, 2.79997519507121699907959137123, 3.72602950562498996867697791057, 5.17423821532097834151764031476, 5.99959557518179057766463581693, 7.52207549186158814310325398101, 8.336799726770287818362657639601, 9.136418024726318708104959741517, 9.606740952203767606460135507511, 10.65467976293229013861474080640

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