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Properties

Label 2-546-91.80-c1-0-7
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: $\chi_{546} (535, \cdot )$
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

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

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