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Properties

Label 2-546-273.128-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.885 - 0.464i$
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 + (0.863 + 1.50i)3-s − 1.00i·4-s + (0.852 + 3.18i)5-s + (−1.67 − 0.450i)6-s + (2.55 + 0.680i)7-s + (0.707 + 0.707i)8-s + (−1.50 + 2.59i)9-s + (−2.85 − 1.64i)10-s + (−4.19 + 1.12i)11-s + (1.50 − 0.863i)12-s + (−3.59 − 0.221i)13-s + (−2.28 + 1.32i)14-s + (−4.03 + 4.02i)15-s − 1.00·16-s + 5.29·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.498 + 0.866i)3-s − 0.500i·4-s + (0.381 + 1.42i)5-s + (−0.682 − 0.184i)6-s + (0.966 + 0.257i)7-s + (0.250 + 0.250i)8-s + (−0.502 + 0.864i)9-s + (−0.902 − 0.520i)10-s + (−1.26 + 0.338i)11-s + (0.433 − 0.249i)12-s + (−0.998 − 0.0615i)13-s + (−0.611 + 0.354i)14-s + (−1.04 + 1.04i)15-s − 0.250·16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.885 - 0.464i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.885 - 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328826 + 1.33532i\)
\(L(\frac12)\) \(\approx\) \(0.328826 + 1.33532i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.863 - 1.50i)T \)
7 \( 1 + (-2.55 - 0.680i)T \)
13 \( 1 + (3.59 + 0.221i)T \)
good5 \( 1 + (-0.852 - 3.18i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (4.19 - 1.12i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 + (-1.57 + 5.88i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 8.97T + 23T^{2} \)
29 \( 1 + (2.35 - 1.36i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.298 - 1.11i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.48 + 1.48i)T - 37iT^{2} \)
41 \( 1 + (8.93 + 2.39i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.42 + 1.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.19 - 1.12i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.94 + 2.85i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.571 - 0.571i)T + 59iT^{2} \)
61 \( 1 + (-1.18 - 2.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.45 + 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.23 - 8.35i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (1.45 + 0.390i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.257 - 0.446i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.56 - 4.56i)T - 83iT^{2} \)
89 \( 1 + (-1.56 - 1.56i)T + 89iT^{2} \)
97 \( 1 + (-2.82 + 0.756i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.82958602409783112432479375517, −10.22903341211315156928136235000, −9.541967255185657740146126231221, −8.509001382816147848470720767132, −7.52624828525556097968812272774, −7.04008664286860975499623555379, −5.31771592306201344022921302584, −5.02871543358913225808622206078, −3.11122293368561655511302992205, −2.33925582324625610528202875489, 0.923539348623528187669261264062, 1.89317113268707109371606333881, 3.24038707021097089283751752805, 4.89708478389235719351694698276, 5.56106242341400227440223456685, 7.30096096770612142503220994651, 8.002429851651080281098524939861, 8.470742738668623261285968617852, 9.490234840262583602323417857927, 10.24344848224168803381089402698

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