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Properties

Label 2-546-21.20-c1-0-5
Degree $2$
Conductor $546$
Sign $0.328 - 0.944i$
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 + (1.72 + 0.146i)3-s − 4-s − 3.83·5-s + (0.146 − 1.72i)6-s + (−0.655 + 2.56i)7-s + i·8-s + (2.95 + 0.507i)9-s + 3.83i·10-s + 3.53i·11-s + (−1.72 − 0.146i)12-s + i·13-s + (2.56 + 0.655i)14-s + (−6.61 − 0.563i)15-s + 16-s − 6.47·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.996 + 0.0848i)3-s − 0.5·4-s − 1.71·5-s + (0.0600 − 0.704i)6-s + (−0.247 + 0.968i)7-s + 0.353i·8-s + (0.985 + 0.169i)9-s + 1.21i·10-s + 1.06i·11-s + (−0.498 − 0.0424i)12-s + 0.277i·13-s + (0.685 + 0.175i)14-s + (−1.70 − 0.145i)15-s + 0.250·16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.328 - 0.944i$
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: \((2,\ 546,\ (\ :1/2),\ 0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795232 + 0.565102i\)
\(L(\frac12)\) \(\approx\) \(0.795232 + 0.565102i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.146i)T \)
7 \( 1 + (0.655 - 2.56i)T \)
13 \( 1 - iT \)
good5 \( 1 + 3.83T + 5T^{2} \)
11 \( 1 - 3.53iT - 11T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 - 1.55iT - 19T^{2} \)
23 \( 1 - 3.87iT - 23T^{2} \)
29 \( 1 + 6.70iT - 29T^{2} \)
31 \( 1 - 9.76iT - 31T^{2} \)
37 \( 1 + 0.597T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 0.604T + 43T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + 6.84iT - 53T^{2} \)
59 \( 1 + 1.74T + 59T^{2} \)
61 \( 1 - 5.28iT - 61T^{2} \)
67 \( 1 - 5.39T + 67T^{2} \)
71 \( 1 - 9.18iT - 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 + 3.29T + 79T^{2} \)
83 \( 1 + 2.06T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−11.16153260235846631728611952257, −9.988605022157002569465044695892, −9.129098719160528822039569407345, −8.469760365557542969749698103859, −7.66728620580276558005531102835, −6.72773813242857430069678718901, −4.80705978058673704405439843232, −4.10723238931391574353756762727, −3.13129805187690871162184423066, −2.01061839671015556681974847708, 0.50211828345514885956757658934, 3.03302143504301649422494088745, 3.98213522139792219635033365806, 4.57192581698980222771445807302, 6.46567371864260446562704933164, 7.21382562303973485641568181815, 7.967608506469034772111156943956, 8.528336920313320932409672269611, 9.393969705210024861885462530347, 10.75717178548855987160679083603

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