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Properties

Label 2-546-39.32-c1-0-9
Degree $2$
Conductor $546$
Sign $0.973 - 0.230i$
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  + (0.965 − 0.258i)2-s + (−1.69 + 0.372i)3-s + (0.866 − 0.499i)4-s + (1.11 + 1.11i)5-s + (−1.53 + 0.797i)6-s + (0.258 − 0.965i)7-s + (0.707 − 0.707i)8-s + (2.72 − 1.26i)9-s + (1.36 + 0.785i)10-s + (1.03 + 3.86i)11-s + (−1.27 + 1.16i)12-s + (−1.97 − 3.01i)13-s − i·14-s + (−2.29 − 1.46i)15-s + (0.500 − 0.866i)16-s + (2.65 + 4.59i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.976 + 0.215i)3-s + (0.433 − 0.249i)4-s + (0.496 + 0.496i)5-s + (−0.627 + 0.325i)6-s + (0.0978 − 0.365i)7-s + (0.249 − 0.249i)8-s + (0.907 − 0.420i)9-s + (0.430 + 0.248i)10-s + (0.311 + 1.16i)11-s + (−0.369 + 0.337i)12-s + (−0.546 − 0.837i)13-s − 0.267i·14-s + (−0.592 − 0.378i)15-s + (0.125 − 0.216i)16-s + (0.643 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.973 - 0.230i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82630 + 0.213555i\)
\(L(\frac12)\) \(\approx\) \(1.82630 + 0.213555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (1.69 - 0.372i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (1.97 + 3.01i)T \)
good5 \( 1 + (-1.11 - 1.11i)T + 5iT^{2} \)
11 \( 1 + (-1.03 - 3.86i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.65 - 4.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.14 - 1.37i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.44 + 2.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.12 - 4.11i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.52 - 3.52i)T - 31iT^{2} \)
37 \( 1 + (-9.27 + 2.48i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.976 - 0.261i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.645 - 0.372i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.72 - 4.72i)T - 47iT^{2} \)
53 \( 1 + 7.57iT - 53T^{2} \)
59 \( 1 + (9.73 + 2.60i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.42 - 2.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.21 + 15.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.21 - 11.9i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.67 + 3.67i)T + 73iT^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + (-2.04 - 2.04i)T + 83iT^{2} \)
89 \( 1 + (4.10 + 15.3i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.62 + 2.04i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.70083426094994384885554682592, −10.24522583862067720218589944188, −9.597756712852110025762683540067, −7.82557924926876767580595715584, −6.92609262418943488504768470550, −6.13508057754587167279775125719, −5.18801640156883112413938271008, −4.36327238043089196234107256075, −3.07878569188869440536518173947, −1.44634811474333968375639927650, 1.20649963928891403561209332351, 2.88416740939119246965873846424, 4.44384859819068055969435264334, 5.33842899552392974127496462971, 5.89629625974790883312953067695, 6.92699977088749265794083091285, 7.80965314850510007106337040402, 9.179255007629739575709053149373, 9.850216211027314035917671230034, 11.25208961885116928742470968259

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