Properties

Label 2-546-91.80-c1-0-5
Degree $2$
Conductor $546$
Sign $0.129 - 0.991i$
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 i·3-s + (−0.866 + 0.499i)4-s + (0.157 − 0.589i)5-s + (0.965 − 0.258i)6-s + (−0.740 + 2.53i)7-s + (−0.707 − 0.707i)8-s − 9-s + 0.610·10-s + (2.49 + 2.49i)11-s + (0.499 + 0.866i)12-s + (1.11 + 3.42i)13-s + (−2.64 − 0.0582i)14-s + (−0.589 − 0.157i)15-s + (0.500 − 0.866i)16-s + (−0.330 − 0.573i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s − 0.577i·3-s + (−0.433 + 0.249i)4-s + (0.0706 − 0.263i)5-s + (0.394 − 0.105i)6-s + (−0.280 + 0.959i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s + 0.192·10-s + (0.751 + 0.751i)11-s + (0.144 + 0.249i)12-s + (0.309 + 0.950i)13-s + (−0.706 − 0.0155i)14-s + (−0.152 − 0.0407i)15-s + (0.125 − 0.216i)16-s + (−0.0802 − 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08865 + 0.955919i\)
\(L(\frac12)\) \(\approx\) \(1.08865 + 0.955919i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (0.740 - 2.53i)T \)
13 \( 1 + (-1.11 - 3.42i)T \)
good5 \( 1 + (-0.157 + 0.589i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.49 - 2.49i)T + 11iT^{2} \)
17 \( 1 + (0.330 + 0.573i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.30 + 1.30i)T + 19iT^{2} \)
23 \( 1 + (-5.75 - 3.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.81 - 4.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.17 - 0.583i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.18 + 0.584i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.83 - 6.84i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.91 + 3.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.42 + 1.18i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.24 + 9.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.32 - 2.49i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (10.1 - 10.1i)T - 67iT^{2} \)
71 \( 1 + (-3.44 - 12.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.08 + 4.06i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.20 + 5.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.17 + 5.17i)T + 83iT^{2} \)
89 \( 1 + (3.90 + 14.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.44 - 0.924i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.33902304675476111486456205598, −9.775628547093418361917807309790, −8.965108756383692343293265100021, −8.497551724888251756977366487926, −6.97372696232067274269266073247, −6.75754668104600845319569096326, −5.52173763334491857658260642562, −4.65083015429429311984668151891, −3.20579060691366140839917273611, −1.66331471287903753837642313542, 0.869918465866741661487758416504, 2.87319521813406055192605165762, 3.72098372970895739607205555511, 4.65029008801781169770134149514, 5.90280473326265653219971435679, 6.81917978347022054076006981913, 8.182360675389983908909200779493, 8.987469525769157900073645519852, 10.03522668191162318226274623491, 10.63577648496025763212185589345

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