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Properties

Label 2-546-91.89-c1-0-2
Degree $2$
Conductor $546$
Sign $0.313 - 0.949i$
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 + (−0.589 − 2.19i)5-s + (−0.258 − 0.965i)6-s + (−0.638 + 2.56i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (1.13 − 1.97i)10-s + (0.513 + 1.91i)11-s + (0.500 − 0.866i)12-s + (0.977 + 3.47i)13-s + (−2.26 + 1.36i)14-s + (−0.589 + 2.19i)15-s − 1.00·16-s + 3.93·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.499 − 0.288i)3-s + 0.500i·4-s + (−0.263 − 0.983i)5-s + (−0.105 − 0.394i)6-s + (−0.241 + 0.970i)7-s + (−0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.359 − 0.623i)10-s + (0.154 + 0.577i)11-s + (0.144 − 0.250i)12-s + (0.271 + 0.962i)13-s + (−0.605 + 0.364i)14-s + (−0.152 + 0.567i)15-s − 0.250·16-s + 0.953·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.313 - 0.949i$
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: \((2,\ 546,\ (\ :1/2),\ 0.313 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19232 + 0.861747i\)
\(L(\frac12)\) \(\approx\) \(1.19232 + 0.861747i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.638 - 2.56i)T \)
13 \( 1 + (-0.977 - 3.47i)T \)
good5 \( 1 + (0.589 + 2.19i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.513 - 1.91i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 3.93T + 17T^{2} \)
19 \( 1 + (-7.46 - 2.00i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 6.41iT - 23T^{2} \)
29 \( 1 + (4.71 + 8.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.57 - 1.22i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (7.60 - 7.60i)T - 37iT^{2} \)
41 \( 1 + (-2.94 - 0.789i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.75 + 1.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.227 - 0.0610i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.68 - 4.64i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.33 + 9.33i)T + 59iT^{2} \)
61 \( 1 + (8.80 - 5.08i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.87 - 1.57i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-11.8 + 3.18i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.54 + 9.50i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.63 - 14.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.22 + 7.22i)T - 83iT^{2} \)
89 \( 1 + (-6.60 - 6.60i)T + 89iT^{2} \)
97 \( 1 + (1.24 + 4.65i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.53982396250629672573826647928, −9.812492721945912716018279340364, −9.216877618425044359147718956583, −8.118909964028083511012736249340, −7.36916899811986241631406944606, −6.19630992167707160060429403817, −5.43176007878710822140390961232, −4.66094881181195412315006779176, −3.36722521222065914655228467640, −1.56657502781970476501066808531, 0.864595678581529813715042627147, 3.13382297915653555675552068622, 3.54866800064462257807712559333, 4.94896253642223423124333742353, 5.89366680663453180926782264743, 6.92992146791423542817580581954, 7.66368733640379849102852124777, 9.137734718610405097363464501227, 10.28582686328448534829474651951, 10.61906847382610802463036624812

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