Properties

Label 2-546-273.38-c1-0-31
Degree $2$
Conductor $546$
Sign $0.963 - 0.268i$
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  + (0.5 + 0.866i)2-s + (1.71 − 0.230i)3-s + (−0.499 + 0.866i)4-s + (1.00 − 0.578i)5-s + (1.05 + 1.37i)6-s + (−0.296 − 2.62i)7-s − 0.999·8-s + (2.89 − 0.790i)9-s + (1.00 + 0.578i)10-s + (1.85 − 3.21i)11-s + (−0.658 + 1.60i)12-s + (−0.361 + 3.58i)13-s + (2.12 − 1.57i)14-s + (1.58 − 1.22i)15-s + (−0.5 − 0.866i)16-s + (0.0128 − 0.0221i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.991 − 0.132i)3-s + (−0.249 + 0.433i)4-s + (0.448 − 0.258i)5-s + (0.431 + 0.559i)6-s + (−0.111 − 0.993i)7-s − 0.353·8-s + (0.964 − 0.263i)9-s + (0.316 + 0.182i)10-s + (0.559 − 0.969i)11-s + (−0.190 + 0.462i)12-s + (−0.100 + 0.994i)13-s + (0.568 − 0.419i)14-s + (0.409 − 0.315i)15-s + (−0.125 − 0.216i)16-s + (0.00310 − 0.00537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.963 - 0.268i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.963 - 0.268i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49274 + 0.340847i\)
\(L(\frac12)\) \(\approx\) \(2.49274 + 0.340847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.71 + 0.230i)T \)
7 \( 1 + (0.296 + 2.62i)T \)
13 \( 1 + (0.361 - 3.58i)T \)
good5 \( 1 + (-1.00 + 0.578i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.85 + 3.21i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.0128 + 0.0221i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.122 + 0.211i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.36 + 1.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.51iT - 29T^{2} \)
31 \( 1 + (2.24 - 3.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.29 - 4.79i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.471iT - 41T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 + (-0.203 + 0.117i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.44 + 3.72i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.80 - 1.61i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.2 + 5.91i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.24 - 3.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (0.784 - 1.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.01 - 5.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.64iT - 83T^{2} \)
89 \( 1 + (-8.11 + 4.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−10.74053866464940835221175340557, −9.656148907417592685114848011494, −8.940768279785773961085991518747, −8.216790234663181250619157374556, −7.03968568244291904746107668114, −6.62335181850494801063074398446, −5.18015235331527737025182899512, −4.04505147103897867028487163052, −3.24250076255127855608848798729, −1.50039811614249370796097142560, 1.88112224763196183895695247197, 2.70298546034262791138114320067, 3.79792459907406588941962570670, 4.98090648818845354560775132823, 6.02745488763870010967228925134, 7.22113221128985975530662302871, 8.310596956117894110619645254321, 9.242580664411177978734405296031, 9.815584266760017003076331997339, 10.56959328834195543685839817172

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