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Properties

Label 2-546-91.6-c1-0-4
Degree $2$
Conductor $546$
Sign $-0.740 - 0.672i$
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  + (0.258 + 0.965i)2-s + (0.866 + 0.5i)3-s + (−0.866 + 0.499i)4-s + (1.52 + 1.52i)5-s + (−0.258 + 0.965i)6-s + (−2.62 − 0.296i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.07 + 1.86i)10-s + (−2.31 + 0.620i)11-s − 12-s + (0.713 + 3.53i)13-s + (−0.394 − 2.61i)14-s + (0.556 + 2.07i)15-s + (0.500 − 0.866i)16-s + (1.44 + 2.49i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.499 + 0.288i)3-s + (−0.433 + 0.249i)4-s + (0.680 + 0.680i)5-s + (−0.105 + 0.394i)6-s + (−0.993 − 0.112i)7-s + (−0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.340 + 0.589i)10-s + (−0.698 + 0.187i)11-s − 0.288·12-s + (0.197 + 0.980i)13-s + (−0.105 − 0.699i)14-s + (0.143 + 0.536i)15-s + (0.125 − 0.216i)16-s + (0.349 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.740 - 0.672i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.740 - 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583102 + 1.50968i\)
\(L(\frac12)\) \(\approx\) \(0.583102 + 1.50968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (2.62 + 0.296i)T \)
13 \( 1 + (-0.713 - 3.53i)T \)
good5 \( 1 + (-1.52 - 1.52i)T + 5iT^{2} \)
11 \( 1 + (2.31 - 0.620i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.44 - 2.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.65 - 6.16i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0337 - 0.0194i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.82 + 6.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.864 - 0.864i)T + 31iT^{2} \)
37 \( 1 + (-5.90 + 1.58i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.54 + 1.21i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.31 + 0.760i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.10 - 3.10i)T - 47iT^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + (-1.12 - 0.301i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.42 + 2.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.334 - 1.25i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.09 + 1.09i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.09 + 9.09i)T - 73iT^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (5.24 + 5.24i)T + 83iT^{2} \)
89 \( 1 + (0.0380 + 0.141i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.46 - 9.21i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.83006098271214929863448581637, −9.972727948343157375639554785986, −9.555013568372296023984458226911, −8.367100956842425634522742495202, −7.54233537002343648674102089540, −6.35682263993218432881884831168, −6.00786236229030480419231415084, −4.46701941868511345030809846076, −3.46373009450722591305557804393, −2.27390802190239167812609137806, 0.841490831281799756128152737216, 2.53835779835837569727836604875, 3.24309132103978441769710359950, 4.80204712375431634541774265734, 5.66141751948660981589651308124, 6.75268367045655626073001818785, 7.998453638919337570240887479723, 8.963703235403591778620253673852, 9.555160755259841651308592586124, 10.36009234570929773671163619139

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