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Properties

Label 2-546-273.251-c1-0-26
Degree $2$
Conductor $546$
Sign $0.350 + 0.936i$
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 + (1.18 − 1.26i)3-s + (−0.499 − 0.866i)4-s − 0.792i·5-s + (0.5 + 1.65i)6-s + (0.5 − 2.59i)7-s + 0.999·8-s + (−0.186 − 2.99i)9-s + (0.686 + 0.396i)10-s + (−2.18 + 3.78i)11-s + (−1.68 − 0.396i)12-s + (3.5 − 0.866i)13-s + (2 + 1.73i)14-s + (−1 − 0.939i)15-s + (−0.5 + 0.866i)16-s + (−2.18 − 3.78i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.684 − 0.728i)3-s + (−0.249 − 0.433i)4-s − 0.354i·5-s + (0.204 + 0.677i)6-s + (0.188 − 0.981i)7-s + 0.353·8-s + (−0.0620 − 0.998i)9-s + (0.216 + 0.125i)10-s + (−0.659 + 1.14i)11-s + (−0.486 − 0.114i)12-s + (0.970 − 0.240i)13-s + (0.534 + 0.462i)14-s + (−0.258 − 0.242i)15-s + (−0.125 + 0.216i)16-s + (−0.530 − 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.350 + 0.936i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12673 - 0.781381i\)
\(L(\frac12)\) \(\approx\) \(1.12673 - 0.781381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 1.26i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 + 0.792iT - 5T^{2} \)
11 \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.18 + 3.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.18 + 2.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.18 + 1.26i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 + (10.1 + 5.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.18 - 4.72i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.939iT - 47T^{2} \)
53 \( 1 + 2.22iT - 53T^{2} \)
59 \( 1 + (-5.31 + 3.06i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.1 - 5.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.05 - 13.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 9.62T + 79T^{2} \)
83 \( 1 - 1.58iT - 83T^{2} \)
89 \( 1 + (-9.30 - 5.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.372 + 0.644i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.42226150759987397819689454546, −9.564817606304717243472454258847, −8.635769782844453361836185708342, −7.911867991805647015733996704220, −7.13599096223703272565421893644, −6.45818511371480079272192802099, −5.01138476228932580053460010461, −3.99720999232823917148758877441, −2.35093333252676357519013681536, −0.851344135118495632976285358205, 1.98555108611611931647074163913, 3.07639802422447340172642941994, 3.92737892537179869299343608314, 5.28337723907797744907851242989, 6.29145799002460928482155558127, 7.938013502616224845201363177331, 8.570726541757673816403099336665, 9.033566501119402696354490860026, 10.29179730426089407640361849458, 10.76159431409444881683089407613

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