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Properties

Label 2-546-273.254-c1-0-4
Degree $2$
Conductor $546$
Sign $0.222 - 0.975i$
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  + (0.965 + 0.258i)2-s + (−1.06 − 1.36i)3-s + (0.866 + 0.499i)4-s + (0.519 + 1.93i)5-s + (−0.675 − 1.59i)6-s + (−2.18 + 1.49i)7-s + (0.707 + 0.707i)8-s + (−0.729 + 2.91i)9-s + 2.00i·10-s + (−0.0346 − 0.0346i)11-s + (−0.240 − 1.71i)12-s + (2.53 + 2.56i)13-s + (−2.49 + 0.881i)14-s + (2.09 − 2.77i)15-s + (0.500 + 0.866i)16-s + (−3.22 + 5.57i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.615 − 0.788i)3-s + (0.433 + 0.249i)4-s + (0.232 + 0.867i)5-s + (−0.275 − 0.651i)6-s + (−0.824 + 0.565i)7-s + (0.249 + 0.249i)8-s + (−0.243 + 0.970i)9-s + 0.634i·10-s + (−0.0104 − 0.0104i)11-s + (−0.0692 − 0.495i)12-s + (0.702 + 0.711i)13-s + (−0.666 + 0.235i)14-s + (0.540 − 0.716i)15-s + (0.125 + 0.216i)16-s + (−0.781 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.222 - 0.975i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.222 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17974 + 0.941142i\)
\(L(\frac12)\) \(\approx\) \(1.17974 + 0.941142i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (1.06 + 1.36i)T \)
7 \( 1 + (2.18 - 1.49i)T \)
13 \( 1 + (-2.53 - 2.56i)T \)
good5 \( 1 + (-0.519 - 1.93i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0346 + 0.0346i)T + 11iT^{2} \)
17 \( 1 + (3.22 - 5.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.694 + 0.694i)T + 19iT^{2} \)
23 \( 1 + (0.854 + 1.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.24 - 0.717i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.76 - 6.57i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-10.9 - 2.93i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.47 + 5.50i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.53 + 5.50i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.68 - 1.79i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.46 - 4.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.37 - 0.369i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 9.47T + 61T^{2} \)
67 \( 1 + (2.68 + 2.68i)T + 67iT^{2} \)
71 \( 1 + (-2.07 - 0.556i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-7.62 - 2.04i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.99 + 3.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (-0.533 + 1.99i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.0946 - 0.353i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.92794775102104906654437323653, −10.62740923470421490872867681810, −9.144515293691052225888369592021, −8.124972108959222013862310793554, −6.87712813341652400648586810040, −6.39505027774492780191654482337, −5.84392646156663002474737225357, −4.42427008027557991566271238253, −3.06549518124996294150445806295, −1.95940459163474290586414075010, 0.75753682822396698431685754652, 2.95217530696372255581324937936, 4.08455677847267612978077190870, 4.85140290178438288533863250551, 5.84219266774493134176724718963, 6.55690303713018789519021120439, 7.892174408973935926474219875811, 9.382003488655531304633257202150, 9.603425095779945944009186682994, 10.85670507827573765330254186435

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