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Properties

Label 2-546-91.83-c1-0-11
Degree $2$
Conductor $546$
Sign $-0.742 + 0.670i$
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 − i·3-s − 1.00i·4-s + (0.864 + 0.864i)5-s + (−0.707 − 0.707i)6-s + (−2.02 − 1.70i)7-s + (−0.707 − 0.707i)8-s − 9-s + 1.22·10-s + (−3.50 − 3.50i)11-s − 1.00·12-s + (3.37 − 1.25i)13-s + (−2.63 + 0.222i)14-s + (0.864 − 0.864i)15-s − 1.00·16-s − 0.322·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.577i·3-s − 0.500i·4-s + (0.386 + 0.386i)5-s + (−0.288 − 0.288i)6-s + (−0.763 − 0.645i)7-s + (−0.250 − 0.250i)8-s − 0.333·9-s + 0.386·10-s + (−1.05 − 1.05i)11-s − 0.288·12-s + (0.937 − 0.348i)13-s + (−0.704 + 0.0593i)14-s + (0.223 − 0.223i)15-s − 0.250·16-s − 0.0781·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.742 + 0.670i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.742 + 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562965 - 1.46399i\)
\(L(\frac12)\) \(\approx\) \(0.562965 - 1.46399i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 + (2.02 + 1.70i)T \)
13 \( 1 + (-3.37 + 1.25i)T \)
good5 \( 1 + (-0.864 - 0.864i)T + 5iT^{2} \)
11 \( 1 + (3.50 + 3.50i)T + 11iT^{2} \)
17 \( 1 + 0.322T + 17T^{2} \)
19 \( 1 + (1.77 + 1.77i)T + 19iT^{2} \)
23 \( 1 + 2.70iT - 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + (-1.72 - 1.72i)T + 31iT^{2} \)
37 \( 1 + (-2.27 - 2.27i)T + 37iT^{2} \)
41 \( 1 + (-6.46 - 6.46i)T + 41iT^{2} \)
43 \( 1 - 0.393iT - 43T^{2} \)
47 \( 1 + (1.41 - 1.41i)T - 47iT^{2} \)
53 \( 1 + 2.03T + 53T^{2} \)
59 \( 1 + (10.7 - 10.7i)T - 59iT^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 + (-5.38 + 5.38i)T - 67iT^{2} \)
71 \( 1 + (0.647 - 0.647i)T - 71iT^{2} \)
73 \( 1 + (-6.85 + 6.85i)T - 73iT^{2} \)
79 \( 1 - 8.81T + 79T^{2} \)
83 \( 1 + (-6.09 - 6.09i)T + 83iT^{2} \)
89 \( 1 + (3.68 - 3.68i)T - 89iT^{2} \)
97 \( 1 + (5.20 + 5.20i)T + 97iT^{2} \)
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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

−10.72084454023475100698666805130, −9.882691907542638113806440437860, −8.654751209179215438601270927274, −7.77062263507861738847494152118, −6.34716997099047584933188873319, −6.18778906516116981101938340571, −4.73274709911052906140753501145, −3.33343598364815961319364089047, −2.59274395589200428529471229923, −0.76481628727888641746598696797, 2.28439175297976300577720139861, 3.56501021195138620335728771643, 4.69007545242350126944335696187, 5.57925422118298538885935462663, 6.32403033080171801586996236551, 7.49975557716244515817479105277, 8.565560621441276613946442012489, 9.350103720896455273517900680514, 10.10923656108037256500120165220, 11.12373974989534580735827057219

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