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Properties

Label 2-546-91.33-c1-0-14
Degree $2$
Conductor $546$
Sign $-0.503 + 0.863i$
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  + (−0.258 + 0.965i)2-s − i·3-s + (−0.866 − 0.499i)4-s + (−0.995 − 3.71i)5-s + (0.965 + 0.258i)6-s + (2.54 − 0.717i)7-s + (0.707 − 0.707i)8-s − 9-s + 3.84·10-s + (−3.32 + 3.32i)11-s + (−0.499 + 0.866i)12-s + (−2.05 − 2.96i)13-s + (0.0336 + 2.64i)14-s + (−3.71 + 0.995i)15-s + (0.500 + 0.866i)16-s + (0.711 − 1.23i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s − 0.577i·3-s + (−0.433 − 0.249i)4-s + (−0.445 − 1.66i)5-s + (0.394 + 0.105i)6-s + (0.962 − 0.271i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s + 1.21·10-s + (−1.00 + 1.00i)11-s + (−0.144 + 0.249i)12-s + (−0.570 − 0.821i)13-s + (0.00899 + 0.707i)14-s + (−0.959 + 0.257i)15-s + (0.125 + 0.216i)16-s + (0.172 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.503 + 0.863i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.503 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412158 - 0.717672i\)
\(L(\frac12)\) \(\approx\) \(0.412158 - 0.717672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + iT \)
7 \( 1 + (-2.54 + 0.717i)T \)
13 \( 1 + (2.05 + 2.96i)T \)
good5 \( 1 + (0.995 + 3.71i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.32 - 3.32i)T - 11iT^{2} \)
17 \( 1 + (-0.711 + 1.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.77 + 1.77i)T - 19iT^{2} \)
23 \( 1 + (7.72 - 4.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.93 + 5.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.79 + 1.01i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.220 - 0.0591i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.90 + 7.10i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.24 + 5.33i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.73 - 2.60i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.442 + 0.766i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.4 + 2.80i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 7.48iT - 61T^{2} \)
67 \( 1 + (-3.51 - 3.51i)T + 67iT^{2} \)
71 \( 1 + (-0.0660 + 0.246i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.34 + 5.01i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.33 + 4.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.32 + 5.32i)T - 83iT^{2} \)
89 \( 1 + (-3.53 + 13.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.63 - 0.705i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.29393714167487326051515902133, −9.437444964672944062124993272324, −8.376197875151278341415752074306, −7.76769054443738628351532914613, −7.38935817513196547885152522983, −5.58239720840006211648425221248, −5.10022520533430680397928985934, −4.17725765139463776030235677532, −1.97057047471814047340511013600, −0.49743945626706343352401127056, 2.24969762931883920827413423013, 3.16935912596368075804278429493, 4.19071392609919444572985488995, 5.39326232259143389224321729458, 6.57621341094309455546671035914, 7.85477771874855145369506033470, 8.305997788455660073550790461862, 9.663991022791144903037717045142, 10.46486378432392142962745442825, 11.00611664865932783452020020112

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