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Properties

Label 2-546-13.10-c1-0-5
Degree $2$
Conductor $546$
Sign $0.967 + 0.252i$
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 + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 3.46i·5-s + (−0.866 − 0.499i)6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.73 + 2.99i)10-s + (3.46 − 2i)11-s − 0.999·12-s + (2.59 + 2.5i)13-s − 0.999·14-s + (2.99 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (3.23 − 5.59i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 1.54i·5-s + (−0.353 − 0.204i)6-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.547 + 0.948i)10-s + (1.04 − 0.603i)11-s − 0.288·12-s + (0.720 + 0.693i)13-s − 0.267·14-s + (0.774 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (0.783 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95640 - 0.251186i\)
\(L(\frac12)\) \(\approx\) \(1.95640 - 0.251186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + (-3.46 + 2i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.23 + 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.46 - 3.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.86 - 4.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4 + 6.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.46iT - 31T^{2} \)
37 \( 1 + (4.26 - 2.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6 - 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.23 + 3.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.535iT - 47T^{2} \)
53 \( 1 + 8.26T + 53T^{2} \)
59 \( 1 + (5.42 + 3.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.59 + 4.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.23 + 3.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.66 - 0.964i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.9iT - 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 + 1.73iT - 83T^{2} \)
89 \( 1 + (10.1 - 5.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.7 - 6.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.15782124276602402605719970348, −10.03269174634094159244269071115, −9.318324256468968088408510378671, −7.69142391746753347386998246056, −6.91557202736186785793798584034, −6.29043475376490839500308371656, −5.33227071454830488358601285270, −3.58100304210979593434574221023, −3.14827916740683880830491384810, −1.45735749955846818859094979571, 1.26226376673888705563999383917, 3.40070455090560381716613569849, 4.31654837945409076956688125335, 5.26073625750644198636961068661, 5.89574126155553846202212122519, 7.09334399883042678238062887757, 8.331701478368411641822987243923, 9.009074783079621196302698514583, 9.790884110916774934292165792518, 10.98435885028860441369972802543

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