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Properties

Label 2-546-91.51-c1-0-14
Degree $2$
Conductor $546$
Sign $0.412 - 0.910i$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (2.38 + 1.37i)5-s + 0.999i·6-s + (0.588 − 2.57i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.37 + 2.38i)10-s + (0.600 − 0.346i)11-s + (−0.499 + 0.866i)12-s + (0.924 + 3.48i)13-s + (1.79 − 1.93i)14-s + 2.74i·15-s + (−0.5 + 0.866i)16-s + (−3.18 − 5.51i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (1.06 + 0.614i)5-s + 0.408i·6-s + (0.222 − 0.974i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.434 + 0.753i)10-s + (0.180 − 0.104i)11-s + (−0.144 + 0.249i)12-s + (0.256 + 0.966i)13-s + (0.480 − 0.518i)14-s + 0.709i·15-s + (−0.125 + 0.216i)16-s + (−0.772 − 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.412 - 0.910i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21318 + 1.42739i\)
\(L(\frac12)\) \(\approx\) \(2.21318 + 1.42739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.588 + 2.57i)T \)
13 \( 1 + (-0.924 - 3.48i)T \)
good5 \( 1 + (-2.38 - 1.37i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.600 + 0.346i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.18 + 5.51i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.78 - 1.02i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.903 + 1.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.85T + 29T^{2} \)
31 \( 1 + (-0.817 + 0.471i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.833 - 0.481i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + 4.97T + 43T^{2} \)
47 \( 1 + (7.81 + 4.51i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.98 - 3.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.5 + 6.06i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.03 + 8.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.40 + 4.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.68iT - 71T^{2} \)
73 \( 1 + (-4.89 + 2.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.339 - 0.587i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.32iT - 83T^{2} \)
89 \( 1 + (11.5 + 6.69i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.89iT - 97T^{2} \)
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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

−11.16243896397238542717937050201, −9.915715531578660296913396041793, −9.452917711165474552298461945309, −8.222800465614419365020648053510, −7.03724532304687935866157747881, −6.51909105331515971065495887178, −5.28619369285800784926131543831, −4.36168369849859305921477979112, −3.28729475812641582605040265472, −2.00849937581288247519591392346, 1.54738700758617357011416827078, 2.40572655532119693815331961337, 3.78037975690768414891117106677, 5.35786586843913858743453657187, 5.68520814451952238724988645115, 6.76865215128790349776029648168, 8.138955546865258678942001474179, 8.941515029236832276496380578866, 9.685954157811659200026184288187, 10.75497612897089363562224711992

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