Properties

Label 2-546-91.16-c1-0-14
Degree $2$
Conductor $546$
Sign $0.460 + 0.887i$
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  + 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.228 + 0.395i)5-s + (0.5 − 0.866i)6-s + (−0.369 − 2.61i)7-s + 8-s + (−0.499 − 0.866i)9-s + (−0.228 + 0.395i)10-s + (1.91 − 3.32i)11-s + (0.5 − 0.866i)12-s + (−3.13 − 1.78i)13-s + (−0.369 − 2.61i)14-s + (0.228 + 0.395i)15-s + 16-s + 1.55·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (−0.102 + 0.176i)5-s + (0.204 − 0.353i)6-s + (−0.139 − 0.990i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0721 + 0.124i)10-s + (0.578 − 1.00i)11-s + (0.144 − 0.249i)12-s + (−0.869 − 0.494i)13-s + (−0.0988 − 0.700i)14-s + (0.0589 + 0.102i)15-s + 0.250·16-s + 0.376·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99111 - 1.21036i\)
\(L(\frac12)\) \(\approx\) \(1.99111 - 1.21036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.369 + 2.61i)T \)
13 \( 1 + (3.13 + 1.78i)T \)
good5 \( 1 + (0.228 - 0.395i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.91 + 3.32i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 + (-1.44 - 2.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.24T + 23T^{2} \)
29 \( 1 + (2.20 + 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.80 - 8.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.280T + 37T^{2} \)
41 \( 1 + (3.57 + 6.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.93 - 6.80i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.550 - 0.953i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.36T + 59T^{2} \)
61 \( 1 + (-5.55 - 9.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.894 + 1.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.06 - 8.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.40 - 2.43i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.70 + 4.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.35T + 83T^{2} \)
89 \( 1 + 0.179T + 89T^{2} \)
97 \( 1 + (-4.73 + 8.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76916855111524583241103288465, −9.945397086530662008837107303227, −8.754945931419918523226661850530, −7.66696845111927181349633700215, −7.06234514150626502589475117593, −6.10132811065521878119733904502, −5.00176765877089241059879115088, −3.69503428115492167533167241869, −2.96011895992169753206188018258, −1.16357820883724181801072898118, 2.09574131163699374491228031648, 3.14721144183630960173121111191, 4.50917239324284242106705905436, 5.04730950327137003606592713419, 6.29789140437920821954041407896, 7.22176572445644725005814393672, 8.344334818092892796801049057025, 9.407647364152835923790715966633, 9.853168595374057002966606545647, 11.17659071458166659190314621609

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