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Properties

Label 2-546-39.8-c1-0-12
Degree $2$
Conductor $546$
Sign $0.683 + 0.729i$
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  + (−0.707 + 0.707i)2-s + (−1.68 − 0.398i)3-s − 1.00i·4-s + (0.559 − 0.559i)5-s + (1.47 − 0.909i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (2.68 + 1.34i)9-s + 0.790i·10-s + (0.347 + 0.347i)11-s + (−0.398 + 1.68i)12-s + (−3.45 − 1.03i)13-s − 1.00i·14-s + (−1.16 + 0.719i)15-s − 1.00·16-s + 4.50·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.973 − 0.230i)3-s − 0.500i·4-s + (0.250 − 0.250i)5-s + (0.601 − 0.371i)6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (0.893 + 0.448i)9-s + 0.250i·10-s + (0.104 + 0.104i)11-s + (−0.115 + 0.486i)12-s + (−0.957 − 0.287i)13-s − 0.267i·14-s + (−0.300 + 0.185i)15-s − 0.250·16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.683 + 0.729i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.683 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632796 - 0.274182i\)
\(L(\frac12)\) \(\approx\) \(0.632796 - 0.274182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.68 + 0.398i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (3.45 + 1.03i)T \)
good5 \( 1 + (-0.559 + 0.559i)T - 5iT^{2} \)
11 \( 1 + (-0.347 - 0.347i)T + 11iT^{2} \)
17 \( 1 - 4.50T + 17T^{2} \)
19 \( 1 + (3.86 + 3.86i)T + 19iT^{2} \)
23 \( 1 - 5.27T + 23T^{2} \)
29 \( 1 + 10.6iT - 29T^{2} \)
31 \( 1 + (2.47 + 2.47i)T + 31iT^{2} \)
37 \( 1 + (-6.92 + 6.92i)T - 37iT^{2} \)
41 \( 1 + (-7.31 + 7.31i)T - 41iT^{2} \)
43 \( 1 + 7.24iT - 43T^{2} \)
47 \( 1 + (-2.49 - 2.49i)T + 47iT^{2} \)
53 \( 1 - 12.0iT - 53T^{2} \)
59 \( 1 + (3.41 + 3.41i)T + 59iT^{2} \)
61 \( 1 - 6.42T + 61T^{2} \)
67 \( 1 + (2.43 + 2.43i)T + 67iT^{2} \)
71 \( 1 + (-8.49 + 8.49i)T - 71iT^{2} \)
73 \( 1 + (-5.93 + 5.93i)T - 73iT^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + (6.31 - 6.31i)T - 83iT^{2} \)
89 \( 1 + (12.2 + 12.2i)T + 89iT^{2} \)
97 \( 1 + (0.867 + 0.867i)T + 97iT^{2} \)
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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

−10.65883208518849236428587362962, −9.695482843781970819131057403031, −9.118837179627021766438724582299, −7.70815540619072920245666635520, −7.16857369543785512853530017071, −6.00752713339257023417098480811, −5.43144493810893297923817939414, −4.34519688725155086084313929241, −2.32364275345863436697611634176, −0.60624934241243740562677162376, 1.26044584569319036350741862623, 2.96772681085531156701607225045, 4.24123578472981185786705356589, 5.30566530494576103356764271485, 6.47177414839529838371406874362, 7.19432879554614750259954554843, 8.344726235624598907914543902593, 9.610553554072187414984099975915, 10.04173860675617552016764020766, 10.85115138027077087501354476983

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