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Properties

Label 2-546-91.33-c1-0-8
Degree $2$
Conductor $546$
Sign $0.571 + 0.820i$
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  + (0.258 − 0.965i)2-s + i·3-s + (−0.866 − 0.499i)4-s + (−0.382 − 1.42i)5-s + (0.965 + 0.258i)6-s + (−1.00 + 2.44i)7-s + (−0.707 + 0.707i)8-s − 9-s − 1.47·10-s + (4.12 − 4.12i)11-s + (0.499 − 0.866i)12-s + (2.68 + 2.41i)13-s + (2.10 + 1.60i)14-s + (1.42 − 0.382i)15-s + (0.500 + 0.866i)16-s + (3.47 − 6.01i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + 0.577i·3-s + (−0.433 − 0.249i)4-s + (−0.171 − 0.638i)5-s + (0.394 + 0.105i)6-s + (−0.381 + 0.924i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s − 0.467·10-s + (1.24 − 1.24i)11-s + (0.144 − 0.249i)12-s + (0.743 + 0.668i)13-s + (0.561 + 0.429i)14-s + (0.368 − 0.0987i)15-s + (0.125 + 0.216i)16-s + (0.842 − 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.571 + 0.820i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.571 + 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35836 - 0.709396i\)
\(L(\frac12)\) \(\approx\) \(1.35836 - 0.709396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 - iT \)
7 \( 1 + (1.00 - 2.44i)T \)
13 \( 1 + (-2.68 - 2.41i)T \)
good5 \( 1 + (0.382 + 1.42i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-4.12 + 4.12i)T - 11iT^{2} \)
17 \( 1 + (-3.47 + 6.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.76 + 3.76i)T - 19iT^{2} \)
23 \( 1 + (-0.666 + 0.384i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.35 - 7.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.379 + 0.101i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.66 + 1.51i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.70 - 6.34i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.142 + 0.0825i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-12.0 + 3.23i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.44 + 7.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.842 + 0.225i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 5.64iT - 61T^{2} \)
67 \( 1 + (3.38 + 3.38i)T + 67iT^{2} \)
71 \( 1 + (1.36 - 5.08i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (2.39 - 8.94i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.96 - 6.86i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.29 + 9.29i)T - 83iT^{2} \)
89 \( 1 + (2.30 - 8.61i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.852 + 0.228i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.89785627873796837190501447069, −9.536553007791141635322840294732, −9.055335512562291662336332383179, −8.595988183433310688816892638139, −6.90458085104463153876421468533, −5.71131251802572070591769849052, −4.99655278211305125472935495137, −3.73053909858273541856632791529, −2.94200708055569673118567878918, −1.04721613452549603278745468977, 1.40182711966378788903529672609, 3.43985453038436666867992088959, 4.08909160473079235990449439122, 5.74399941418789880845336582393, 6.46209396264617247140399161383, 7.38468462694199568956782191626, 7.79661612013261516682117387763, 9.091386796197829744564418564863, 10.07469273948661318554869369258, 10.82020319144410853914371942164

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