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Properties

Label 2-546-273.128-c1-0-7
Degree $2$
Conductor $546$
Sign $0.0627 - 0.998i$
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 + (0.717 + 1.57i)3-s − 1.00i·4-s + (0.882 + 3.29i)5-s + (1.62 + 0.607i)6-s + (−2.36 + 1.18i)7-s + (−0.707 − 0.707i)8-s + (−1.96 + 2.26i)9-s + (2.95 + 1.70i)10-s + (−2.62 + 0.704i)11-s + (1.57 − 0.717i)12-s + (0.963 − 3.47i)13-s + (−0.835 + 2.51i)14-s + (−4.55 + 3.75i)15-s − 1.00·16-s + 5.38·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.414 + 0.910i)3-s − 0.500i·4-s + (0.394 + 1.47i)5-s + (0.662 + 0.247i)6-s + (−0.894 + 0.447i)7-s + (−0.250 − 0.250i)8-s + (−0.656 + 0.754i)9-s + (0.933 + 0.539i)10-s + (−0.792 + 0.212i)11-s + (0.455 − 0.207i)12-s + (0.267 − 0.963i)13-s + (−0.223 + 0.670i)14-s + (−1.17 + 0.969i)15-s − 0.250·16-s + 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0627 - 0.998i$
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: \((2,\ 546,\ (\ :1/2),\ 0.0627 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36321 + 1.28015i\)
\(L(\frac12)\) \(\approx\) \(1.36321 + 1.28015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.717 - 1.57i)T \)
7 \( 1 + (2.36 - 1.18i)T \)
13 \( 1 + (-0.963 + 3.47i)T \)
good5 \( 1 + (-0.882 - 3.29i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.62 - 0.704i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 5.38T + 17T^{2} \)
19 \( 1 + (1.50 - 5.63i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 + (-7.19 + 4.15i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.302 + 1.12i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.35 - 3.35i)T - 37iT^{2} \)
41 \( 1 + (-1.36 - 0.366i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.98 - 5.18i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-12.4 + 3.34i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.31 + 1.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.350 + 0.350i)T + 59iT^{2} \)
61 \( 1 + (-6.12 - 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.546 - 0.146i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.45 - 9.16i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (13.2 + 3.53i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.696 - 1.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.65 + 2.65i)T - 83iT^{2} \)
89 \( 1 + (9.70 + 9.70i)T + 89iT^{2} \)
97 \( 1 + (-12.8 + 3.44i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.61033685411255249397400051275, −10.16286663248179379773269393043, −9.892267673481739676585455605336, −8.452684904681742171860799576323, −7.43056597866230070152551947250, −5.98024629705701827967648739241, −5.63803919901951024914773249521, −3.99902315157725792019874586489, −3.03640176388744264589777284925, −2.55835897549011906600336543369, 0.890690037406920705913730865527, 2.57399018109276624745467231732, 3.90740287194213669582836473135, 5.12003016360530043662271639350, 6.01060884211834219719933994952, 6.94310545283340147199256521831, 7.80917427631279201755656186600, 8.835658430056179461744967454146, 9.234180746940816161923545253315, 10.55705233913226594140567598532

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