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Properties

Label 2-546-273.185-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.327 - 0.944i$
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 + (−1.59 + 0.672i)3-s + (0.499 + 0.866i)4-s + (1.07 + 1.86i)5-s + (1.71 + 0.215i)6-s + (0.447 + 2.60i)7-s − 0.999i·8-s + (2.09 − 2.14i)9-s − 2.15i·10-s − 0.832i·11-s + (−1.38 − 1.04i)12-s + (1.46 + 3.29i)13-s + (0.916 − 2.48i)14-s + (−2.96 − 2.24i)15-s + (−0.5 + 0.866i)16-s + (1.36 + 2.35i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.921 + 0.388i)3-s + (0.249 + 0.433i)4-s + (0.480 + 0.832i)5-s + (0.701 + 0.0879i)6-s + (0.169 + 0.985i)7-s − 0.353i·8-s + (0.698 − 0.715i)9-s − 0.679i·10-s − 0.250i·11-s + (−0.398 − 0.301i)12-s + (0.405 + 0.914i)13-s + (0.244 − 0.663i)14-s + (−0.766 − 0.580i)15-s + (−0.125 + 0.216i)16-s + (0.330 + 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.327 - 0.944i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.327 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.432305 + 0.607446i\)
\(L(\frac12)\) \(\approx\) \(0.432305 + 0.607446i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.672i)T \)
7 \( 1 + (-0.447 - 2.60i)T \)
13 \( 1 + (-1.46 - 3.29i)T \)
good5 \( 1 + (-1.07 - 1.86i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.832iT - 11T^{2} \)
17 \( 1 + (-1.36 - 2.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 2.56iT - 19T^{2} \)
23 \( 1 + (2.15 + 1.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.62 - 4.40i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.06 + 1.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0899 + 0.155i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.93 - 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.89 - 6.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.425 - 0.736i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 5.99i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.26 - 2.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.65iT - 61T^{2} \)
67 \( 1 + 6.61T + 67T^{2} \)
71 \( 1 + (7.48 + 4.32i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (10.9 + 6.31i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.10 + 5.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.930T + 83T^{2} \)
89 \( 1 + (-0.572 + 0.992i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.08 - 4.09i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.09815408280064389928640507866, −10.28260381699342925718150122644, −9.428495182230906701533394977154, −8.745321984960856349940019572749, −7.40509213115965568749259009803, −6.34909991853891837358216491336, −5.81422684474370053000155078126, −4.43237793318797098213534403002, −3.08161972642723844931689131941, −1.72454388512031237180816814387, 0.61297041462971875411848514855, 1.75753436549296184688493582830, 4.01548999914674526269491061502, 5.34306305275688115730634277147, 5.77476620518884934671241353990, 7.12570313338932961197082244466, 7.63971620396545789580294149559, 8.703815678079392529955819920874, 9.846641172049149743697756000331, 10.38534052928135510239362162243

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