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Properties

Label 2-546-91.54-c1-0-5
Degree $2$
Conductor $546$
Sign $0.679 - 0.733i$
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 + (−3.71 + 0.995i)5-s + (−0.965 + 0.258i)6-s + (1.84 − 1.89i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (1.92 − 3.33i)10-s + (4.54 − 1.21i)11-s + (0.500 − 0.866i)12-s + (2.05 − 2.96i)13-s + (0.0336 + 2.64i)14-s + (−3.71 − 0.995i)15-s − 1.00·16-s + 1.42·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.499 + 0.288i)3-s − 0.500i·4-s + (−1.66 + 0.445i)5-s + (−0.394 + 0.105i)6-s + (0.698 − 0.716i)7-s + (0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.608 − 1.05i)10-s + (1.36 − 0.366i)11-s + (0.144 − 0.250i)12-s + (0.570 − 0.821i)13-s + (0.00899 + 0.707i)14-s + (−0.959 − 0.257i)15-s − 0.250·16-s + 0.345·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07677 + 0.470393i\)
\(L(\frac12)\) \(\approx\) \(1.07677 + 0.470393i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 + 1.89i)T \)
13 \( 1 + (-2.05 + 2.96i)T \)
good5 \( 1 + (3.71 - 0.995i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-4.54 + 1.21i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 1.42T + 17T^{2} \)
19 \( 1 + (0.651 - 2.42i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 8.92iT - 23T^{2} \)
29 \( 1 + (-2.93 - 5.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.01 - 3.79i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.161 + 0.161i)T + 37iT^{2} \)
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 \( 1 + (2.60 + 9.73i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.442 + 0.766i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.65 + 7.65i)T - 59iT^{2} \)
61 \( 1 + (-6.48 + 3.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.28 - 4.79i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.0660 - 0.246i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.01 - 1.34i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-2.33 + 4.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.32 + 5.32i)T + 83iT^{2} \)
89 \( 1 + (9.66 - 9.66i)T - 89iT^{2} \)
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

−10.98203365867811545831537232667, −10.03717913181421859009361646895, −8.837754554599807464999505147768, −8.174889587110517270230256502861, −7.51656180961900097634544772738, −6.78664460158366720821520972848, −5.30956434169369923640317332559, −3.91024657088890238406164525577, −3.53005353195062337396072449445, −1.15724826301940209656382509804, 1.06479845508402115975800116456, 2.54868545354346040556748322982, 4.05960031932871141243988055458, 4.43719331437256966271534742698, 6.40595290593310745404985477515, 7.40112277030445418935979945335, 8.315361201203085812222980237957, 8.722299472017492970916048715184, 9.508225963648936700961499666155, 11.02676670592635717160983580703

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