Properties

Label 2-546-39.8-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.350 - 0.936i$
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 + (−0.337 − 1.69i)3-s − 1.00i·4-s + (−2.19 + 2.19i)5-s + (−1.44 − 0.962i)6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−2.77 + 1.14i)9-s + 3.09i·10-s + (−1.27 − 1.27i)11-s + (−1.69 + 0.337i)12-s + (−3.23 + 1.58i)13-s + 1.00i·14-s + (4.46 + 2.98i)15-s − 1.00·16-s − 3.68·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.194 − 0.980i)3-s − 0.500i·4-s + (−0.980 + 0.980i)5-s + (−0.587 − 0.392i)6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.923 + 0.382i)9-s + 0.980i·10-s + (−0.384 − 0.384i)11-s + (−0.490 + 0.0974i)12-s + (−0.897 + 0.440i)13-s + 0.267i·14-s + (1.15 + 0.770i)15-s − 0.250·16-s − 0.892·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.350 - 0.936i$
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: \((2,\ 546,\ (\ :1/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0567890 + 0.0819269i\)
\(L(\frac12)\) \(\approx\) \(0.0567890 + 0.0819269i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.337 + 1.69i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (3.23 - 1.58i)T \)
good5 \( 1 + (2.19 - 2.19i)T - 5iT^{2} \)
11 \( 1 + (1.27 + 1.27i)T + 11iT^{2} \)
17 \( 1 + 3.68T + 17T^{2} \)
19 \( 1 + (-5.26 - 5.26i)T + 19iT^{2} \)
23 \( 1 + 8.15T + 23T^{2} \)
29 \( 1 + 9.50iT - 29T^{2} \)
31 \( 1 + (0.125 + 0.125i)T + 31iT^{2} \)
37 \( 1 + (0.328 - 0.328i)T - 37iT^{2} \)
41 \( 1 + (-2.21 + 2.21i)T - 41iT^{2} \)
43 \( 1 - 1.43iT - 43T^{2} \)
47 \( 1 + (-0.805 - 0.805i)T + 47iT^{2} \)
53 \( 1 - 5.59iT - 53T^{2} \)
59 \( 1 + (1.49 + 1.49i)T + 59iT^{2} \)
61 \( 1 + 8.14T + 61T^{2} \)
67 \( 1 + (10.6 + 10.6i)T + 67iT^{2} \)
71 \( 1 + (-0.752 + 0.752i)T - 71iT^{2} \)
73 \( 1 + (-3.59 + 3.59i)T - 73iT^{2} \)
79 \( 1 - 4.38T + 79T^{2} \)
83 \( 1 + (9.35 - 9.35i)T - 83iT^{2} \)
89 \( 1 + (3.19 + 3.19i)T + 89iT^{2} \)
97 \( 1 + (-1.79 - 1.79i)T + 97iT^{2} \)
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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

−11.37334652102969797224041207677, −10.49173834579224930251632786778, −9.490071815675092671552830556261, −8.016198232108143872102180299454, −7.52775447379401091382557849538, −6.44765418419195266704513721397, −5.71221928027130598156818187642, −4.23778702322780054221689767596, −3.09722577460946485309866484410, −2.13009656458528767253806291517, 0.04668420959033085122899467663, 2.96728391141206837128661041287, 4.14061680503295037814566873847, 4.78937256287907114901486032223, 5.51245269118544233729650780362, 6.95223167708808746032772555885, 7.82990388510280118606300145943, 8.752489351449916351471592749269, 9.540093268873712961997189574614, 10.54438954394053028382384080017

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