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Properties

Label 2-546-273.32-c1-0-32
Degree $2$
Conductor $546$
Sign $-0.998 + 0.0581i$
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  + (−0.707 − 0.707i)2-s + (−0.745 + 1.56i)3-s + 1.00i·4-s + (1.03 − 3.87i)5-s + (1.63 − 0.577i)6-s + (−1.61 − 2.09i)7-s + (0.707 − 0.707i)8-s + (−1.88 − 2.33i)9-s + (−3.47 + 2.00i)10-s + (1.57 + 0.421i)11-s + (−1.56 − 0.745i)12-s + (−2.67 + 2.41i)13-s + (−0.340 + 2.62i)14-s + (5.27 + 4.50i)15-s − 1.00·16-s − 5.88·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.430 + 0.902i)3-s + 0.500i·4-s + (0.463 − 1.73i)5-s + (0.666 − 0.235i)6-s + (−0.610 − 0.792i)7-s + (0.250 − 0.250i)8-s + (−0.629 − 0.777i)9-s + (−1.09 + 0.633i)10-s + (0.474 + 0.127i)11-s + (−0.451 − 0.215i)12-s + (−0.742 + 0.670i)13-s + (−0.0910 + 0.701i)14-s + (1.36 + 1.16i)15-s − 0.250·16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.998 + 0.0581i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.998 + 0.0581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00996511 - 0.342337i\)
\(L(\frac12)\) \(\approx\) \(0.00996511 - 0.342337i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.745 - 1.56i)T \)
7 \( 1 + (1.61 + 2.09i)T \)
13 \( 1 + (2.67 - 2.41i)T \)
good5 \( 1 + (-1.03 + 3.87i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.57 - 0.421i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 5.88T + 17T^{2} \)
19 \( 1 + (-1.73 - 6.46i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.97T + 23T^{2} \)
29 \( 1 + (-2.46 - 1.42i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.90 + 7.10i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.59 + 5.59i)T + 37iT^{2} \)
41 \( 1 + (2.15 - 0.577i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.83 + 2.79i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.99 + 0.802i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.23 + 3.02i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.53 - 2.53i)T - 59iT^{2} \)
61 \( 1 + (-2.04 + 3.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.2 - 2.75i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.37 + 5.14i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.06 + 0.554i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.573 - 0.993i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.681 + 0.681i)T + 83iT^{2} \)
89 \( 1 + (7.35 - 7.35i)T - 89iT^{2} \)
97 \( 1 + (5.15 + 1.38i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.04978486342468345400543319557, −9.602084404884628039842721280924, −9.004540982630634730858109979017, −8.042691786743615629169140181988, −6.62306936535423786386995601972, −5.54681019100904850881926820917, −4.37682437632756386198549737825, −3.91722980744414621256573791389, −1.84918834336719291132171250311, −0.22955499407944947359034108330, 2.18380031922710440275377627406, 2.97024825937934125651102326264, 5.15451229172744140310527277654, 6.25910685518505806603447494478, 6.64907741999226198519189197471, 7.31411164020742359869729969346, 8.473046994683199956846233475722, 9.496196276168996410058756621466, 10.37590152325122850231448289446, 11.14500566736185485443208078171

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