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Properties

Label 2-546-91.89-c1-0-9
Degree $2$
Conductor $546$
Sign $0.245 - 0.969i$
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 + (0.866 + 0.5i)3-s + 1.00i·4-s + (−0.367 − 1.37i)5-s + (0.258 + 0.965i)6-s + (1.50 + 2.17i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.710 − 1.23i)10-s + (0.998 + 3.72i)11-s + (−0.500 + 0.866i)12-s + (1.74 − 3.15i)13-s + (−0.469 + 2.60i)14-s + (0.367 − 1.37i)15-s − 1.00·16-s + 2.36·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.499 + 0.288i)3-s + 0.500i·4-s + (−0.164 − 0.614i)5-s + (0.105 + 0.394i)6-s + (0.570 + 0.821i)7-s + (−0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.224 − 0.389i)10-s + (0.301 + 1.12i)11-s + (−0.144 + 0.250i)12-s + (0.484 − 0.875i)13-s + (−0.125 + 0.695i)14-s + (0.0949 − 0.354i)15-s − 0.250·16-s + 0.572·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.245 - 0.969i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.245 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79299 + 1.39541i\)
\(L(\frac12)\) \(\approx\) \(1.79299 + 1.39541i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-1.50 - 2.17i)T \)
13 \( 1 + (-1.74 + 3.15i)T \)
good5 \( 1 + (0.367 + 1.37i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.998 - 3.72i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 2.36T + 17T^{2} \)
19 \( 1 + (5.84 + 1.56i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 8.10iT - 23T^{2} \)
29 \( 1 + (0.220 + 0.382i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.19 - 0.855i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-5.36 + 5.36i)T - 37iT^{2} \)
41 \( 1 + (-1.33 - 0.357i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.53 + 4.35i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.14 - 1.64i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.08 + 7.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.93 + 5.93i)T + 59iT^{2} \)
61 \( 1 + (-9.22 + 5.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0380 + 0.0101i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (12.2 - 3.28i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.06 + 7.72i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.14 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.37 + 9.37i)T - 83iT^{2} \)
89 \( 1 + (-1.45 - 1.45i)T + 89iT^{2} \)
97 \( 1 + (3.62 + 13.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.11218963695076309894442379705, −9.917534980545739111450895919951, −9.018870948098465233415118845950, −8.264546037685017409074285103283, −7.59097886631213218066567316487, −6.30495722867296530941222797383, −5.21142242656712346186948465581, −4.54723598931786645438218828659, −3.35438701984683902305590673604, −1.92307642499437928867498370527, 1.25257654347692524814611842651, 2.71604090986732725672247947865, 3.78256586592446273023451853580, 4.59532286251585503670900683002, 6.23577737636818328398347572472, 6.76980035579991187972986759747, 8.084864846667611241284468105191, 8.701529280899855111582759077256, 9.993835978289510334154986211199, 10.83701114902404771242526985122

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