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Properties

Label 2-546-91.51-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.720 - 0.693i$
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 + (1.87 + 1.08i)5-s + 0.999i·6-s + (−2.59 − 0.5i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.08 − 1.87i)10-s + (−3.60 + 2.08i)11-s + (0.499 − 0.866i)12-s + (−0.418 + 3.58i)13-s + (2 + 1.73i)14-s − 2.16i·15-s + (−0.5 + 0.866i)16-s + (−2.58 − 4.47i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (0.837 + 0.483i)5-s + 0.408i·6-s + (−0.981 − 0.188i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.341 − 0.592i)10-s + (−1.08 + 0.627i)11-s + (0.144 − 0.249i)12-s + (−0.116 + 0.993i)13-s + (0.534 + 0.462i)14-s − 0.558i·15-s + (−0.125 + 0.216i)16-s + (−0.626 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.720 - 0.693i$
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: \((2,\ 546,\ (\ :1/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0492936 + 0.122210i\)
\(L(\frac12)\) \(\approx\) \(0.0492936 + 0.122210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 0.5i)T \)
13 \( 1 + (0.418 - 3.58i)T \)
good5 \( 1 + (-1.87 - 1.08i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.60 - 2.08i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.58 + 4.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.47 + 3.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.581 - 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
31 \( 1 + (7.79 - 4.5i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.47 + 3.16i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 9.16T + 43T^{2} \)
47 \( 1 + (-11.6 - 6.74i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.08 - 5.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.59 - 0.918i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.74 - 13.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.45 + 1.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.67iT - 71T^{2} \)
73 \( 1 + (-5.47 + 3.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.8iT - 83T^{2} \)
89 \( 1 + (-2.73 - 1.58i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11iT - 97T^{2} \)
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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

−10.81954303647014979795067239064, −10.36590844705141354819145786631, −9.436178535344619711743132536817, −8.741623030988008942764405443591, −7.18583919943183221471736355879, −6.93321570816718146077626957616, −5.87256336032866656979531958829, −4.51876772984389874798193446389, −2.80993142323714348717819491412, −2.04698060623114900890581401757, 0.086135528698739362817174202080, 2.17789977756980157092939959578, 3.62381954540945084292457976923, 5.23331316972792301627954380784, 5.84853539612981470554891972320, 6.60363782918757374885651114732, 8.092220720711743031515046322496, 8.713542949166273703236415104984, 9.656826789791514727235692972256, 10.43687059043916932953497057935

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