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Properties

Label 2-546-21.20-c1-0-19
Degree $2$
Conductor $546$
Sign $0.962 + 0.270i$
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 + (1.52 + 0.814i)3-s − 4-s + 1.76·5-s + (0.814 − 1.52i)6-s + (2.58 − 0.566i)7-s + i·8-s + (1.67 + 2.48i)9-s − 1.76i·10-s + 3.38i·11-s + (−1.52 − 0.814i)12-s − i·13-s + (−0.566 − 2.58i)14-s + (2.69 + 1.43i)15-s + 16-s − 5.58·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.882 + 0.470i)3-s − 0.5·4-s + 0.787·5-s + (0.332 − 0.624i)6-s + (0.976 − 0.213i)7-s + 0.353i·8-s + (0.557 + 0.829i)9-s − 0.556i·10-s + 1.02i·11-s + (−0.441 − 0.235i)12-s − 0.277i·13-s + (−0.151 − 0.690i)14-s + (0.694 + 0.370i)15-s + 0.250·16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.962 + 0.270i$
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: \((2,\ 546,\ (\ :1/2),\ 0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20126 - 0.303367i\)
\(L(\frac12)\) \(\approx\) \(2.20126 - 0.303367i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 0.814i)T \)
7 \( 1 + (-2.58 + 0.566i)T \)
13 \( 1 + iT \)
good5 \( 1 - 1.76T + 5T^{2} \)
11 \( 1 - 3.38iT - 11T^{2} \)
17 \( 1 + 5.58T + 17T^{2} \)
19 \( 1 - 3.39iT - 19T^{2} \)
23 \( 1 + 3.58iT - 23T^{2} \)
29 \( 1 + 3.65iT - 29T^{2} \)
31 \( 1 + 9.08iT - 31T^{2} \)
37 \( 1 - 9.49T + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 - 7.79T + 43T^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 - 4.99iT - 53T^{2} \)
59 \( 1 + 0.204T + 59T^{2} \)
61 \( 1 + 3.18iT - 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + 5.92iT - 71T^{2} \)
73 \( 1 + 8.84iT - 73T^{2} \)
79 \( 1 - 1.31T + 79T^{2} \)
83 \( 1 - 0.878T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 5.82iT - 97T^{2} \)
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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

−10.62148409192013445953457992220, −9.853490263669682519291011437207, −9.268667943877569473231715793557, −8.237307474257882056987029792692, −7.52531240684418577210909591474, −6.00179066079558172311264162921, −4.65339554743731365871276356017, −4.16617628959739997132378678017, −2.49049894480705249185775081187, −1.82673748426074576247765543446, 1.50822197525519734464006711256, 2.80145335477734691535068727482, 4.27121053977593911810463374213, 5.40496718661698289563269446607, 6.41510719352954582406131601833, 7.23969812226472012794793488463, 8.305016367638972438128394305203, 8.840196424888726428345140881671, 9.541930587291639253640008550014, 10.82393052693381763736397167180

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