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Properties

Label 2-546-273.152-c1-0-14
Degree $2$
Conductor $546$
Sign $0.982 - 0.185i$
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  + (−0.866 + 0.5i)2-s + (0.696 − 1.58i)3-s + (0.499 − 0.866i)4-s + (−1.42 + 2.47i)5-s + (0.189 + 1.72i)6-s + (2.54 − 0.718i)7-s + 0.999i·8-s + (−2.02 − 2.20i)9-s − 2.85i·10-s + 2.06i·11-s + (−1.02 − 1.39i)12-s + (3.41 + 1.15i)13-s + (−1.84 + 1.89i)14-s + (2.92 + 3.98i)15-s + (−0.5 − 0.866i)16-s + (2.04 − 3.53i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.402 − 0.915i)3-s + (0.249 − 0.433i)4-s + (−0.637 + 1.10i)5-s + (0.0773 + 0.702i)6-s + (0.962 − 0.271i)7-s + 0.353i·8-s + (−0.676 − 0.736i)9-s − 0.902i·10-s + 0.621i·11-s + (−0.295 − 0.403i)12-s + (0.947 + 0.320i)13-s + (−0.493 + 0.506i)14-s + (0.754 + 1.02i)15-s + (−0.125 − 0.216i)16-s + (0.495 − 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.982 - 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25243 + 0.117016i\)
\(L(\frac12)\) \(\approx\) \(1.25243 + 0.117016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.696 + 1.58i)T \)
7 \( 1 + (-2.54 + 0.718i)T \)
13 \( 1 + (-3.41 - 1.15i)T \)
good5 \( 1 + (1.42 - 2.47i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 2.06iT - 11T^{2} \)
17 \( 1 + (-2.04 + 3.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 4.34iT - 19T^{2} \)
23 \( 1 + (-1.19 + 0.689i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.75 - 3.90i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.44 + 1.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.17 - 5.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.55 + 6.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.02 + 8.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.65 + 6.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.80 + 3.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.42 - 9.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.94iT - 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (5.66 - 3.27i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.16 + 1.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.08 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (3.01 + 5.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (15.8 - 9.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.82566209911972131270114961690, −9.993033128802267271209889718618, −8.660197140854098840581466451717, −8.103999477110826859749550734031, −7.15946708334203444646698652965, −6.84573222572524682736428602468, −5.54369095814644607267819266401, −3.94634825088080717630584824719, −2.65292610751097472670749519432, −1.27779024268941280083572980981, 1.10531156715477575970387894988, 2.86523531258889675884991191494, 4.09992943455698518759324192909, 4.84945992178560594282668472897, 6.01147632808210156187462591296, 7.896843446622776206344124984554, 8.313892753081775463173091452614, 8.845786677307888290820183077910, 9.775421881114090548886975613823, 10.96052909956015011641216919772

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