Properties

Label 2-546-273.272-c1-0-20
Degree $2$
Conductor $546$
Sign $0.229 + 0.973i$
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 + (−1.26 − 1.18i)3-s + 4-s − 3.37i·5-s + (1.26 + 1.18i)6-s + (2.52 + 0.792i)7-s − 8-s + (0.186 + 2.99i)9-s + 3.37i·10-s + 5.74·11-s + (−1.26 − 1.18i)12-s + (3.46 + i)13-s + (−2.52 − 0.792i)14-s + (−4 + 4.25i)15-s + 16-s + 0.792·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.728 − 0.684i)3-s + 0.5·4-s − 1.50i·5-s + (0.515 + 0.484i)6-s + (0.954 + 0.299i)7-s − 0.353·8-s + (0.0620 + 0.998i)9-s + 1.06i·10-s + 1.73·11-s + (−0.364 − 0.342i)12-s + (0.960 + 0.277i)13-s + (−0.674 − 0.211i)14-s + (−1.03 + 1.09i)15-s + 0.250·16-s + 0.192·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.229 + 0.973i$
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: \((2,\ 546,\ (\ :1/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823279 - 0.651919i\)
\(L(\frac12)\) \(\approx\) \(0.823279 - 0.651919i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 1.18i)T \)
7 \( 1 + (-2.52 - 0.792i)T \)
13 \( 1 + (-3.46 - i)T \)
good5 \( 1 + 3.37iT - 5T^{2} \)
11 \( 1 - 5.74T + 11T^{2} \)
17 \( 1 - 0.792T + 17T^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 - 0.147iT - 23T^{2} \)
29 \( 1 - 2.37iT - 29T^{2} \)
31 \( 1 + 4.10T + 31T^{2} \)
37 \( 1 + 8.36iT - 37T^{2} \)
41 \( 1 - 8.37iT - 41T^{2} \)
43 \( 1 + 2.62T + 43T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 2.74iT - 59T^{2} \)
61 \( 1 + 7.74iT - 61T^{2} \)
67 \( 1 - 5.98iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 + 6.74iT - 83T^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + 9.15T + 97T^{2} \)
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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

−10.84739055767760673578614552102, −9.421751424972474743613055596006, −8.798731809368889132936541452519, −8.126391221806260117895144730119, −7.08877105423016673498236459616, −6.02195554123601111082938107354, −5.19800897871101185171250105243, −4.07801534114046681141860787923, −1.66495823251427008282235398559, −1.11004299824629714803495968392, 1.37665565081363843645827531542, 3.27329465912881978846715112476, 4.15574509224355772173576616921, 5.70503695722635526173934667176, 6.55565438621278119700156698161, 7.22311082880176453688814512406, 8.435544694119958868753722769059, 9.435235756775393684502882567371, 10.25254392724090404748680118157, 11.00030900134179790700218404393

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