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Properties

Label 2-546-21.20-c1-0-2
Degree $2$
Conductor $546$
Sign $0.999 + 0.00810i$
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  − i·2-s + (−1.59 − 0.684i)3-s − 4-s − 4.42·5-s + (−0.684 + 1.59i)6-s + (−2.43 + 1.02i)7-s + i·8-s + (2.06 + 2.17i)9-s + 4.42i·10-s − 5.78i·11-s + (1.59 + 0.684i)12-s − i·13-s + (1.02 + 2.43i)14-s + (7.03 + 3.02i)15-s + 16-s + 2.97·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.918 − 0.395i)3-s − 0.5·4-s − 1.97·5-s + (−0.279 + 0.649i)6-s + (−0.921 + 0.387i)7-s + 0.353i·8-s + (0.687 + 0.726i)9-s + 1.39i·10-s − 1.74i·11-s + (0.459 + 0.197i)12-s − 0.277i·13-s + (0.274 + 0.651i)14-s + (1.81 + 0.781i)15-s + 0.250·16-s + 0.721·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.999 + 0.00810i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.999 + 0.00810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356351 - 0.00144423i\)
\(L(\frac12)\) \(\approx\) \(0.356351 - 0.00144423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.59 + 0.684i)T \)
7 \( 1 + (2.43 - 1.02i)T \)
13 \( 1 + iT \)
good5 \( 1 + 4.42T + 5T^{2} \)
11 \( 1 + 5.78iT - 11T^{2} \)
17 \( 1 - 2.97T + 17T^{2} \)
19 \( 1 - 2.66iT - 19T^{2} \)
23 \( 1 - 6.57iT - 23T^{2} \)
29 \( 1 - 3.44iT - 29T^{2} \)
31 \( 1 + 3.06iT - 31T^{2} \)
37 \( 1 + 4.71T + 37T^{2} \)
41 \( 1 + 2.07T + 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 - 4.28T + 47T^{2} \)
53 \( 1 - 5.36iT - 53T^{2} \)
59 \( 1 + 2.04T + 59T^{2} \)
61 \( 1 - 7.83iT - 61T^{2} \)
67 \( 1 + 2.80T + 67T^{2} \)
71 \( 1 + 4.46iT - 71T^{2} \)
73 \( 1 - 4.11iT - 73T^{2} \)
79 \( 1 - 1.88T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 5.79iT - 97T^{2} \)
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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

−11.00667659841712311922158148584, −10.34289301596752780252634128543, −8.978389738881064582047922794166, −8.082479556500878614529095772993, −7.34421206700334602205319953665, −6.10216251137126047874785404207, −5.19075224702998452798738878511, −3.73477007545274128910280133999, −3.23622444377220913552819422208, −0.842460331746639550472967705475, 0.35592316101486831087156671112, 3.47631605181378882009710632555, 4.36389998501334792973516050783, 4.93065448096619975402243407726, 6.61409236460839793660144868019, 7.01395510858611730316017616616, 7.82699593122299105829470796441, 9.011789655121165223345149209830, 10.01935905864394218565546088853, 10.71160506477284198595753155929

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