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Properties

Label 2-546-91.47-c1-0-7
Degree $2$
Conductor $546$
Sign $0.994 - 0.107i$
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  + (−0.965 + 0.258i)2-s + (−0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (0.0181 + 0.0678i)5-s + (0.707 − 0.707i)6-s + (2.49 − 0.873i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.0351 − 0.0608i)10-s + (−0.570 − 0.152i)11-s + (−0.5 + 0.866i)12-s + (1.58 + 3.24i)13-s + (−2.18 + 1.48i)14-s + (−0.0496 − 0.0496i)15-s + (0.500 − 0.866i)16-s + (−2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (0.00813 + 0.0303i)5-s + (0.288 − 0.288i)6-s + (0.943 − 0.330i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.0111 − 0.0192i)10-s + (−0.171 − 0.0460i)11-s + (−0.144 + 0.249i)12-s + (0.438 + 0.898i)13-s + (−0.584 + 0.398i)14-s + (−0.0128 − 0.0128i)15-s + (0.125 − 0.216i)16-s + (−0.635 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.994 - 0.107i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.994 - 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994734 + 0.0537414i\)
\(L(\frac12)\) \(\approx\) \(0.994734 + 0.0537414i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.49 + 0.873i)T \)
13 \( 1 + (-1.58 - 3.24i)T \)
good5 \( 1 + (-0.0181 - 0.0678i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.570 + 0.152i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.62 + 4.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.51 + 5.64i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.01 - 2.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.18T + 29T^{2} \)
31 \( 1 + (-8.24 - 2.20i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.83 - 6.86i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.36 - 3.36i)T - 41iT^{2} \)
43 \( 1 - 1.42iT - 43T^{2} \)
47 \( 1 + (-10.8 + 2.91i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.67 - 2.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.13 + 4.22i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.23 - 0.713i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.63 + 9.82i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.04 - 8.04i)T + 71iT^{2} \)
73 \( 1 + (-2.94 + 11.0i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.725 - 1.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.15 - 7.15i)T - 83iT^{2} \)
89 \( 1 + (16.1 - 4.33i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (9.94 - 9.94i)T - 97iT^{2} \)
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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

−11.01462562328308883617232856595, −9.930067247967804896221761107408, −9.002469900672307344786216597352, −8.325612730042374347263896797669, −7.06049162642033878640174984959, −6.57567387020598329444194564116, −5.08408895060124216646799428509, −4.47407049247659773715571479355, −2.65639359377780561360218270778, −1.00473875853256142179935520296, 1.16113806504790760263437227084, 2.45532088662424273388221953525, 4.08453195422981319888540883456, 5.38321088725741784351608166296, 6.22263302594034404227175484479, 7.35921949481199230603406006945, 8.295202370097996301647256450652, 8.741117072329146268358414547946, 10.20703213055046538326266977703, 10.73133606925752962877986111822

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