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Properties

Label 2-546-273.2-c1-0-30
Degree $2$
Conductor $546$
Sign $-0.805 - 0.592i$
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 + (−1.72 − 0.143i)3-s + 1.00i·4-s + (−1.09 + 0.293i)5-s + (1.11 + 1.32i)6-s + (−0.0234 − 2.64i)7-s + (0.707 − 0.707i)8-s + (2.95 + 0.493i)9-s + (0.983 + 0.567i)10-s + (−0.882 − 3.29i)11-s + (0.143 − 1.72i)12-s + (2.35 + 2.73i)13-s + (−1.85 + 1.88i)14-s + (1.93 − 0.350i)15-s − 1.00·16-s + 2.19·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.996 − 0.0825i)3-s + 0.500i·4-s + (−0.490 + 0.131i)5-s + (0.457 + 0.539i)6-s + (−0.00885 − 0.999i)7-s + (0.250 − 0.250i)8-s + (0.986 + 0.164i)9-s + (0.310 + 0.179i)10-s + (−0.266 − 0.992i)11-s + (0.0412 − 0.498i)12-s + (0.652 + 0.757i)13-s + (−0.495 + 0.504i)14-s + (0.499 − 0.0904i)15-s − 0.250·16-s + 0.533·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0285874 + 0.0871231i\)
\(L(\frac12)\) \(\approx\) \(0.0285874 + 0.0871231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 + 0.143i)T \)
7 \( 1 + (0.0234 + 2.64i)T \)
13 \( 1 + (-2.35 - 2.73i)T \)
good5 \( 1 + (1.09 - 0.293i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.882 + 3.29i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 2.19T + 17T^{2} \)
19 \( 1 + (2.74 + 0.735i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 8.10T + 23T^{2} \)
29 \( 1 + (4.77 - 2.75i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.48 - 1.46i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (8.14 + 8.14i)T + 37iT^{2} \)
41 \( 1 + (1.60 - 5.97i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.18 + 3.56i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.43 - 9.09i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.67 - 0.966i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.878 - 0.878i)T - 59iT^{2} \)
61 \( 1 + (-1.71 - 2.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.31 - 8.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.82 + 1.29i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-4.09 + 15.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.69 - 4.67i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.78 + 9.78i)T + 83iT^{2} \)
89 \( 1 + (11.4 - 11.4i)T - 89iT^{2} \)
97 \( 1 + (1.01 + 3.77i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.51774123185893638526586535874, −9.684541582568748832158892414126, −8.422917600542888004488754405794, −7.61101730755885601711924160496, −6.70200015045790003477051398428, −5.73364525991298613134506360134, −4.28486388473197066446355463577, −3.57355801893209632711374248083, −1.55631184643369289170217477418, −0.07205138600573298707296468434, 1.91121231701786876300340042694, 3.91913220727761551826504765498, 5.11237053135795684700527130683, 5.86557389493459253903976753712, 6.69786593223983127203237160112, 7.889707843280670561460564334915, 8.433442929911385606319946266527, 9.891144421157320666321322874992, 10.13719368527613661500453624128, 11.39022312928919924771632950561

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