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Properties

Label 2-546-39.8-c1-0-8
Degree $2$
Conductor $546$
Sign $0.111 - 0.993i$
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 + (1.71 + 0.264i)3-s − 1.00i·4-s + (−0.790 + 0.790i)5-s + (−1.39 + 1.02i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (2.85 + 0.906i)9-s − 1.11i·10-s + (3.45 + 3.45i)11-s + (0.264 − 1.71i)12-s + (−1.18 − 3.40i)13-s − 1.00i·14-s + (−1.56 + 1.14i)15-s − 1.00·16-s − 0.401·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.988 + 0.152i)3-s − 0.500i·4-s + (−0.353 + 0.353i)5-s + (−0.570 + 0.417i)6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (0.953 + 0.302i)9-s − 0.353i·10-s + (1.04 + 1.04i)11-s + (0.0764 − 0.494i)12-s + (−0.329 − 0.944i)13-s − 0.267i·14-s + (−0.403 + 0.295i)15-s − 0.250·16-s − 0.0972·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.111 - 0.993i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.111 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11010 + 0.992444i\)
\(L(\frac12)\) \(\approx\) \(1.11010 + 0.992444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.264i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (1.18 + 3.40i)T \)
good5 \( 1 + (0.790 - 0.790i)T - 5iT^{2} \)
11 \( 1 + (-3.45 - 3.45i)T + 11iT^{2} \)
17 \( 1 + 0.401T + 17T^{2} \)
19 \( 1 + (-4.88 - 4.88i)T + 19iT^{2} \)
23 \( 1 + 6.12T + 23T^{2} \)
29 \( 1 + 1.16iT - 29T^{2} \)
31 \( 1 + (-2.88 - 2.88i)T + 31iT^{2} \)
37 \( 1 + (4.48 - 4.48i)T - 37iT^{2} \)
41 \( 1 + (-6.76 + 6.76i)T - 41iT^{2} \)
43 \( 1 + 2.74iT - 43T^{2} \)
47 \( 1 + (-5.67 - 5.67i)T + 47iT^{2} \)
53 \( 1 + 3.46iT - 53T^{2} \)
59 \( 1 + (8.03 + 8.03i)T + 59iT^{2} \)
61 \( 1 + 0.717T + 61T^{2} \)
67 \( 1 + (0.693 + 0.693i)T + 67iT^{2} \)
71 \( 1 + (-5.15 + 5.15i)T - 71iT^{2} \)
73 \( 1 + (0.0455 - 0.0455i)T - 73iT^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + (1.74 - 1.74i)T - 83iT^{2} \)
89 \( 1 + (-8.55 - 8.55i)T + 89iT^{2} \)
97 \( 1 + (7.03 + 7.03i)T + 97iT^{2} \)
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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

−10.62210820386381392351806239510, −9.803975915120906561297890338146, −9.350868758153426276662703878347, −8.213564207447484808787173360470, −7.57346011092867137435511394100, −6.80681393605309889632027249473, −5.56401640937352746718423366619, −4.21716829415660037291471839635, −3.18252348203362596834007011178, −1.72269581565415926303688644097, 1.01310560106938058739205629878, 2.51008675729502416590278260488, 3.68138815149513446032578571368, 4.43348908881636070357153244502, 6.29774568645917695146045356365, 7.25896114179176608336858976990, 8.111161720009081493903069591277, 9.057291366437276188187231262676, 9.378278038725586424000070869061, 10.45658723414015981857536190703

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