Properties

Label 2-546-39.8-c1-0-20
Degree $2$
Conductor $546$
Sign $0.239 + 0.971i$
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 + (1.15 − 1.29i)3-s − 1.00i·4-s + (0.616 − 0.616i)5-s + (0.0958 + 1.72i)6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.331 − 2.98i)9-s + 0.872i·10-s + (−2.63 − 2.63i)11-s + (−1.29 − 1.15i)12-s + (1.08 − 3.43i)13-s + 1.00i·14-s + (−0.0836 − 1.50i)15-s − 1.00·16-s − 5.36·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.666 − 0.745i)3-s − 0.500i·4-s + (0.275 − 0.275i)5-s + (0.0391 + 0.706i)6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.110 − 0.993i)9-s + 0.275i·10-s + (−0.793 − 0.793i)11-s + (−0.372 − 0.333i)12-s + (0.300 − 0.953i)13-s + 0.267i·14-s + (−0.0215 − 0.389i)15-s − 0.250·16-s − 1.30·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.239 + 0.971i$
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: \((2,\ 546,\ (\ :1/2),\ 0.239 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05010 - 0.822951i\)
\(L(\frac12)\) \(\approx\) \(1.05010 - 0.822951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 + 1.29i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-1.08 + 3.43i)T \)
good5 \( 1 + (-0.616 + 0.616i)T - 5iT^{2} \)
11 \( 1 + (2.63 + 2.63i)T + 11iT^{2} \)
17 \( 1 + 5.36T + 17T^{2} \)
19 \( 1 + (-3.60 - 3.60i)T + 19iT^{2} \)
23 \( 1 - 2.67T + 23T^{2} \)
29 \( 1 - 3.00iT - 29T^{2} \)
31 \( 1 + (6.60 + 6.60i)T + 31iT^{2} \)
37 \( 1 + (-4.44 + 4.44i)T - 37iT^{2} \)
41 \( 1 + (-8.43 + 8.43i)T - 41iT^{2} \)
43 \( 1 + 0.610iT - 43T^{2} \)
47 \( 1 + (-3.46 - 3.46i)T + 47iT^{2} \)
53 \( 1 - 0.464iT - 53T^{2} \)
59 \( 1 + (-8.93 - 8.93i)T + 59iT^{2} \)
61 \( 1 + 8.37T + 61T^{2} \)
67 \( 1 + (-1.77 - 1.77i)T + 67iT^{2} \)
71 \( 1 + (0.00980 - 0.00980i)T - 71iT^{2} \)
73 \( 1 + (-11.1 + 11.1i)T - 73iT^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 + (3.29 - 3.29i)T - 83iT^{2} \)
89 \( 1 + (-8.90 - 8.90i)T + 89iT^{2} \)
97 \( 1 + (5.41 + 5.41i)T + 97iT^{2} \)
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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

−10.62105732337916407365370320027, −9.331015074712709659737009129114, −8.820409914094011563469917102553, −7.78191044339802057075987228783, −7.40327788925210884138580009330, −6.06675392388215935811635238330, −5.37594809148990858531109895280, −3.69295183430930119394483898496, −2.33245740099804142203600963811, −0.853258555836415953098221298151, 2.04720580251779214286725564249, 2.85172969219578143731868924427, 4.27136432929459096935850333185, 5.04264416884544036285826906477, 6.66217587943788132003156455518, 7.64137541515448533998881798451, 8.631881042874069946131760673005, 9.313332316465626317442451128241, 9.970755930724341880864972485019, 10.96345061772333167894515203246

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