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Properties

Label 2-546-21.20-c1-0-20
Degree $2$
Conductor $546$
Sign $0.693 - 0.720i$
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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.12601 + 0.904422i\)
\(L(\frac12)\) \(\approx\) \(2.12601 + 0.904422i\)
\(L(\frac{3}{2})\) 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

−10.16863153424200676026775588507, −9.906340281657679146810429468273, −9.184244023406120164743946881250, −8.392466548465372393969924171758, −6.98015055492624540903580428883, −6.57433953652247767970585354613, −5.62453313658762661373702948827, −4.39812481825891421823072431489, −2.70083902407770049367680535672, −1.87641972468384453349681771771, 1.52826153776323574319309850129, 2.78755445690264734795008357779, 3.46935100501053697695460981992, 5.00601673853349970075356113560, 5.98399570800883309523382055604, 6.95997848072249449313324933566, 8.613259233297300472680466249889, 9.129640923207919336463488073144, 9.752795346886929619074694754015, 10.50933251140227790779001326521

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