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Properties

Label 2-546-273.32-c1-0-18
Degree $2$
Conductor $546$
Sign $0.588 - 0.808i$
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 + (0.958 − 1.44i)3-s + 1.00i·4-s + (−0.421 + 1.57i)5-s + (1.69 − 0.342i)6-s + (−1.51 + 2.17i)7-s + (−0.707 + 0.707i)8-s + (−1.16 − 2.76i)9-s + (−1.40 + 0.813i)10-s + (4.90 + 1.31i)11-s + (1.44 + 0.958i)12-s + (3.58 + 0.426i)13-s + (−2.60 + 0.467i)14-s + (1.86 + 2.11i)15-s − 1.00·16-s + 0.853·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.553 − 0.833i)3-s + 0.500i·4-s + (−0.188 + 0.702i)5-s + (0.693 − 0.139i)6-s + (−0.570 + 0.820i)7-s + (−0.250 + 0.250i)8-s + (−0.388 − 0.921i)9-s + (−0.445 + 0.257i)10-s + (1.47 + 0.396i)11-s + (0.416 + 0.276i)12-s + (0.992 + 0.118i)13-s + (−0.695 + 0.124i)14-s + (0.481 + 0.545i)15-s − 0.250·16-s + 0.206·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.588 - 0.808i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93679 + 0.986125i\)
\(L(\frac12)\) \(\approx\) \(1.93679 + 0.986125i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.958 + 1.44i)T \)
7 \( 1 + (1.51 - 2.17i)T \)
13 \( 1 + (-3.58 - 0.426i)T \)
good5 \( 1 + (0.421 - 1.57i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.90 - 1.31i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.853T + 17T^{2} \)
19 \( 1 + (-1.44 - 5.41i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 2.49T + 23T^{2} \)
29 \( 1 + (4.98 + 2.88i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.47 + 5.51i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.88 - 3.88i)T + 37iT^{2} \)
41 \( 1 + (2.07 - 0.555i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-11.2 + 6.49i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.23 + 1.93i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (9.96 + 5.75i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.583 + 0.583i)T - 59iT^{2} \)
61 \( 1 + (2.92 - 5.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.8 - 3.17i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.90 + 14.5i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (10.5 - 2.83i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.03 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.18 + 3.18i)T + 83iT^{2} \)
89 \( 1 + (-5.02 + 5.02i)T - 89iT^{2} \)
97 \( 1 + (14.9 + 4.00i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.35410795540097076859986233497, −9.753479082099553668313159676460, −9.027688901778827768725661748736, −8.093816576564132618430162528918, −7.20707424600052483054880384217, −6.29793140473072196968620582549, −5.90563434909490552663947782693, −3.92897622497178169877918143365, −3.26464810965036192608219986362, −1.84621464484990113821178390425, 1.17777583347748557155358326274, 3.11442497944237863775382140069, 3.90585182126229763891133511112, 4.60857128422936775910835673548, 5.85148430829931225048280944256, 6.95715730098904124993573614962, 8.301324430893281867081506580598, 9.202487523027197870153653289083, 9.618381316083442951557271302320, 10.95716789089556427739295137894

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