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Properties

Label 2-546-273.173-c1-0-33
Degree $2$
Conductor $546$
Sign $-0.688 + 0.724i$
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  + (0.5 + 0.866i)2-s + (−1.51 − 0.844i)3-s + (−0.499 + 0.866i)4-s + (−0.511 − 0.295i)5-s + (−0.0248 − 1.73i)6-s + (1.57 − 2.12i)7-s − 0.999·8-s + (1.57 + 2.55i)9-s − 0.590i·10-s − 6.11·11-s + (1.48 − 0.887i)12-s + (−1.86 + 3.08i)13-s + (2.62 + 0.304i)14-s + (0.523 + 0.877i)15-s + (−0.5 − 0.866i)16-s + (3.82 − 6.62i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.873 − 0.487i)3-s + (−0.249 + 0.433i)4-s + (−0.228 − 0.131i)5-s + (−0.0101 − 0.707i)6-s + (0.596 − 0.802i)7-s − 0.353·8-s + (0.524 + 0.851i)9-s − 0.186i·10-s − 1.84·11-s + (0.429 − 0.256i)12-s + (−0.518 + 0.855i)13-s + (0.702 + 0.0814i)14-s + (0.135 + 0.226i)15-s + (−0.125 − 0.216i)16-s + (0.927 − 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.688 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.688 + 0.724i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.688 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118316 - 0.275684i\)
\(L(\frac12)\) \(\approx\) \(0.118316 - 0.275684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.51 + 0.844i)T \)
7 \( 1 + (-1.57 + 2.12i)T \)
13 \( 1 + (1.86 - 3.08i)T \)
good5 \( 1 + (0.511 + 0.295i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + 6.11T + 11T^{2} \)
17 \( 1 + (-3.82 + 6.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.95T + 19T^{2} \)
23 \( 1 + (7.26 - 4.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.39 + 0.803i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.38 + 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.93 - 1.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.36 + 0.787i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.90 + 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.94 - 2.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.30 + 1.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.88 + 2.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.100iT - 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + (-0.875 - 1.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.41 - 9.37i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.17 - 5.50i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.07iT - 83T^{2} \)
89 \( 1 + (-3.56 + 2.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.82 - 4.89i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.51247493349419782267096236422, −9.812969579026273082542154560351, −8.157008660652575278021727870866, −7.61790740615404508782219676815, −6.99331840888557754585540394044, −5.69723517847890624339366832109, −5.01707547306724315481654396165, −4.13177108342390460156230138697, −2.25005738060268765802292717277, −0.16063283359892337533772151763, 2.05735120348050958120661472124, 3.41773865392706262499937440668, 4.64281711334923870242418763541, 5.49529411434902339206846890091, 6.03394098329733460100032688939, 7.73292973657496042372226500327, 8.426850142147319395763915129972, 9.824756441034670161971694595534, 10.60517734747901072431463205692, 10.82629625813807797869979785206

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