Properties

Label 2-546-273.272-c1-0-19
Degree $2$
Conductor $546$
Sign $-0.453 + 0.891i$
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  − 2-s + (−1.5 − 0.866i)3-s + 4-s + 1.73i·5-s + (1.5 + 0.866i)6-s + (−2.5 + 0.866i)7-s − 8-s + (1.5 + 2.59i)9-s − 1.73i·10-s + (−1.5 − 0.866i)12-s + (1 − 3.46i)13-s + (2.5 − 0.866i)14-s + (1.49 − 2.59i)15-s + 16-s − 3·17-s + (−1.5 − 2.59i)18-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.866 − 0.499i)3-s + 0.5·4-s + 0.774i·5-s + (0.612 + 0.353i)6-s + (−0.944 + 0.327i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s − 0.547i·10-s + (−0.433 − 0.249i)12-s + (0.277 − 0.960i)13-s + (0.668 − 0.231i)14-s + (0.387 − 0.670i)15-s + 0.250·16-s − 0.727·17-s + (−0.353 − 0.612i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.453 + 0.891i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.453 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176029 - 0.287228i\)
\(L(\frac12)\) \(\approx\) \(0.176029 - 0.287228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
13 \( 1 + (-1 + 3.46i)T \)
good5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + 12.1iT - 47T^{2} \)
53 \( 1 + 3.46iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 13.8iT - 67T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 3.46iT - 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.47940537339231293610537384530, −9.902753679929592513909221530771, −8.726364576074737566685452204941, −7.69830171852438734904832294612, −6.77010883631986819751188178852, −6.26677459256349196366371373804, −5.21843143127830986396278176720, −3.42246719177726680751014629698, −2.21039341179798672450867367705, −0.28739572070691110036812848507, 1.34458609380992478097970620333, 3.41251745273826628381294344249, 4.53724676716970795690871960480, 5.63466075184820213684113710937, 6.62202373195864845150671672461, 7.33273285142611350748768584801, 8.967462412349409188519111699868, 9.184848695926766363294939809326, 10.16811071041008149835523994473, 10.98744514629988858298932280200

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