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Properties

Label 2-546-39.8-c1-0-16
Degree $2$
Conductor $546$
Sign $0.739 + 0.672i$
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 + (1.41 − i)3-s − 1.00i·4-s + (2.47 − 2.47i)5-s + (−0.292 + 1.70i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (1.00 − 2.82i)9-s + 3.49i·10-s + (−0.976 − 0.976i)11-s + (−1.00 − 1.41i)12-s + (1.76 + 3.14i)13-s − 1.00i·14-s + (1.02 − 5.96i)15-s − 1.00·16-s + 5.49·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.816 − 0.577i)3-s − 0.500i·4-s + (1.10 − 1.10i)5-s + (−0.119 + 0.696i)6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (0.333 − 0.942i)9-s + 1.10i·10-s + (−0.294 − 0.294i)11-s + (−0.288 − 0.408i)12-s + (0.489 + 0.872i)13-s − 0.267i·14-s + (0.264 − 1.54i)15-s − 0.250·16-s + 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.739 + 0.672i$
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: \((2,\ 546,\ (\ :1/2),\ 0.739 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61009 - 0.622827i\)
\(L(\frac12)\) \(\approx\) \(1.61009 - 0.622827i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-1.76 - 3.14i)T \)
good5 \( 1 + (-2.47 + 2.47i)T - 5iT^{2} \)
11 \( 1 + (0.976 + 0.976i)T + 11iT^{2} \)
17 \( 1 - 5.49T + 17T^{2} \)
19 \( 1 + (-0.470 - 0.470i)T + 19iT^{2} \)
23 \( 1 + 8.97T + 23T^{2} \)
29 \( 1 + 2.03iT - 29T^{2} \)
31 \( 1 + (-0.585 - 0.585i)T + 31iT^{2} \)
37 \( 1 + (1.56 - 1.56i)T - 37iT^{2} \)
41 \( 1 + (0.966 - 0.966i)T - 41iT^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + (4.24 + 4.24i)T + 47iT^{2} \)
53 \( 1 - 13.9iT - 53T^{2} \)
59 \( 1 + (-8.81 - 8.81i)T + 59iT^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 + (-4.36 - 4.36i)T + 67iT^{2} \)
71 \( 1 + (0.908 - 0.908i)T - 71iT^{2} \)
73 \( 1 + (11.0 - 11.0i)T - 73iT^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 + (6.49 - 6.49i)T - 83iT^{2} \)
89 \( 1 + (1.74 + 1.74i)T + 89iT^{2} \)
97 \( 1 + (-5.41 - 5.41i)T + 97iT^{2} \)
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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

−10.05180272088318982052198497049, −9.739546801098819008294520538892, −8.688652228102113865714304556022, −8.359836817331667510574508110348, −7.20078409795164268044732370781, −6.07703904005182826940052444611, −5.51332741351191584478194136748, −3.95285670525458624105898585748, −2.26633390247342216878810543248, −1.21909870915730345018136940142, 1.89213655908833083363052665752, 2.95756790492504096588887497499, 3.69373247651219246018946455813, 5.32383222306265845434732202748, 6.42447348355494156379902998119, 7.62482485224162728631207070143, 8.266938815576180605037684009096, 9.602124027028432283043418904776, 10.02875366560158771091773885928, 10.43351160892368714742860488316

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