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Properties

Label 2-546-91.89-c1-0-13
Degree $2$
Conductor $546$
Sign $0.848 + 0.529i$
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 + (−0.866 − 0.5i)3-s + 1.00i·4-s + (−0.0285 − 0.106i)5-s + (−0.258 − 0.965i)6-s + (0.253 − 2.63i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.0551 − 0.0955i)10-s + (−1.22 − 4.58i)11-s + (0.500 − 0.866i)12-s + (3.58 + 0.403i)13-s + (2.04 − 1.68i)14-s + (−0.0285 + 0.106i)15-s − 1.00·16-s + 2.90·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.499 − 0.288i)3-s + 0.500i·4-s + (−0.0127 − 0.0476i)5-s + (−0.105 − 0.394i)6-s + (0.0958 − 0.995i)7-s + (−0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.0174 − 0.0302i)10-s + (−0.370 − 1.38i)11-s + (0.144 − 0.250i)12-s + (0.993 + 0.111i)13-s + (0.545 − 0.449i)14-s + (−0.00737 + 0.0275i)15-s − 0.250·16-s + 0.704·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.848 + 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49287 - 0.427771i\)
\(L(\frac12)\) \(\approx\) \(1.49287 - 0.427771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.253 + 2.63i)T \)
13 \( 1 + (-3.58 - 0.403i)T \)
good5 \( 1 + (0.0285 + 0.106i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.22 + 4.58i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 2.90T + 17T^{2} \)
19 \( 1 + (2.58 + 0.692i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.34iT - 23T^{2} \)
29 \( 1 + (0.975 + 1.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.39 - 2.51i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.39 - 5.39i)T - 37iT^{2} \)
41 \( 1 + (-6.25 - 1.67i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.18 + 1.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.89 - 1.58i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.68 + 6.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.23 - 2.23i)T + 59iT^{2} \)
61 \( 1 + (-6.85 + 3.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.83 + 2.10i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.73 - 1.53i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.38 - 5.15i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.92 - 3.32i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.03 - 8.03i)T - 83iT^{2} \)
89 \( 1 + (10.0 + 10.0i)T + 89iT^{2} \)
97 \( 1 + (-3.07 - 11.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.86171729993332210587170169132, −10.10814105452427809608033214906, −8.430769300294879246381945268219, −8.166020548088731291241645309349, −6.75131917716965573790745742569, −6.30975862614258639812114802669, −5.17954418408487680144371365787, −4.16958778687495746949248819860, −3.03162598931001793629809880932, −0.892752710612980708793149787857, 1.66622126322335749762963187065, 3.03305433296196821031033297806, 4.28237426237314004816767485271, 5.25983332129941602569588858869, 5.97131770004790497182259087474, 7.09573906285261958406857540647, 8.341709721028194777429288547302, 9.393775195333797195772391782523, 10.11422875864717897335424786832, 11.00272197823550106955001458621

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