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Properties

Label 2-546-91.25-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.809 - 0.586i$
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 + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−2.38 + 1.37i)5-s + 0.999i·6-s + (−0.588 − 2.57i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (1.37 − 2.38i)10-s + (−0.600 − 0.346i)11-s + (−0.499 − 0.866i)12-s + (0.924 + 3.48i)13-s + (1.79 + 1.93i)14-s + 2.74i·15-s + (−0.5 − 0.866i)16-s + (−3.18 + 5.51i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−1.06 + 0.614i)5-s + 0.408i·6-s + (−0.222 − 0.974i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.434 − 0.753i)10-s + (−0.180 − 0.104i)11-s + (−0.144 − 0.249i)12-s + (0.256 + 0.966i)13-s + (0.480 + 0.518i)14-s + 0.709i·15-s + (−0.125 − 0.216i)16-s + (−0.772 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.809 - 0.586i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.809 - 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0913093 + 0.281599i\)
\(L(\frac12)\) \(\approx\) \(0.0913093 + 0.281599i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.588 + 2.57i)T \)
13 \( 1 + (-0.924 - 3.48i)T \)
good5 \( 1 + (2.38 - 1.37i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.600 + 0.346i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.18 - 5.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.78 - 1.02i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.903 - 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.85T + 29T^{2} \)
31 \( 1 + (0.817 + 0.471i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.833 - 0.481i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + 4.97T + 43T^{2} \)
47 \( 1 + (-7.81 + 4.51i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.98 + 3.43i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.5 + 6.06i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.03 - 8.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.40 + 4.85i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.68iT - 71T^{2} \)
73 \( 1 + (4.89 + 2.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.339 + 0.587i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.32iT - 83T^{2} \)
89 \( 1 + (-11.5 + 6.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.89iT - 97T^{2} \)
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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

−11.08594301515875898496652670546, −10.36146843703570591219533764642, −9.220447496016031914223413456567, −8.297575359499543210643118669488, −7.55259332709468791748130957952, −6.90308958506846915948112291009, −6.11417395760673853516831404460, −4.29010236417195234565524686161, −3.45671095224864430200780731026, −1.73542502308649536687931666438, 0.19483960100297875299586249542, 2.39676978301935963765291517638, 3.47966254063721733182296356003, 4.62155860327301388752488171088, 5.65730092769042826471609058482, 7.19120443553907102430502370843, 8.022001905999477157550693246188, 8.916412431365211477303279582431, 9.237600510387383937869172312163, 10.50014127732156476907914148299

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