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Properties

Label 2-546-39.8-c1-0-4
Degree $2$
Conductor $546$
Sign $0.974 - 0.222i$
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 + (−1.41 − i)3-s − 1.00i·4-s + (−2.23 + 2.23i)5-s + (−1.70 + 0.292i)6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (1.00 + 2.82i)9-s + 3.15i·10-s + (3.38 + 3.38i)11-s + (−1.00 + 1.41i)12-s + (−1.52 + 3.26i)13-s − 1.00i·14-s + (5.38 − 0.924i)15-s − 1.00·16-s + 5.15·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.816 − 0.577i)3-s − 0.500i·4-s + (−0.998 + 0.998i)5-s + (−0.696 + 0.119i)6-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (0.333 + 0.942i)9-s + 0.998i·10-s + (1.02 + 1.02i)11-s + (−0.288 + 0.408i)12-s + (−0.422 + 0.906i)13-s − 0.267i·14-s + (1.39 − 0.238i)15-s − 0.250·16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.974 - 0.222i$
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: \((2,\ 546,\ (\ :1/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16809 + 0.131754i\)
\(L(\frac12)\) \(\approx\) \(1.16809 + 0.131754i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (1.52 - 3.26i)T \)
good5 \( 1 + (2.23 - 2.23i)T - 5iT^{2} \)
11 \( 1 + (-3.38 - 3.38i)T + 11iT^{2} \)
17 \( 1 - 5.15T + 17T^{2} \)
19 \( 1 + (4.23 + 4.23i)T + 19iT^{2} \)
23 \( 1 - 6.67T + 23T^{2} \)
29 \( 1 - 4.20iT - 29T^{2} \)
31 \( 1 + (-3.41 - 3.41i)T + 31iT^{2} \)
37 \( 1 + (0.0256 - 0.0256i)T - 37iT^{2} \)
41 \( 1 + (7.20 - 7.20i)T - 41iT^{2} \)
43 \( 1 - 8.08iT - 43T^{2} \)
47 \( 1 + (-4.24 - 4.24i)T + 47iT^{2} \)
53 \( 1 - 2.87iT - 53T^{2} \)
59 \( 1 + (-2.48 - 2.48i)T + 59iT^{2} \)
61 \( 1 - 2.47T + 61T^{2} \)
67 \( 1 + (7.19 + 7.19i)T + 67iT^{2} \)
71 \( 1 + (-2.25 + 2.25i)T - 71iT^{2} \)
73 \( 1 + (-8.14 + 8.14i)T - 73iT^{2} \)
79 \( 1 - 2.36T + 79T^{2} \)
83 \( 1 + (6.15 - 6.15i)T - 83iT^{2} \)
89 \( 1 + (9.55 + 9.55i)T + 89iT^{2} \)
97 \( 1 + (-2.58 - 2.58i)T + 97iT^{2} \)
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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

−11.06497080576373673281666156308, −10.39868769010701171797219740684, −9.273628633653271228700655869579, −7.80630053184397003053411712553, −6.85884956458887754855076682645, −6.64052815593442368833331620590, −4.95115295055113521302569875152, −4.27083944301256594047612502730, −2.94034709727898106800483848449, −1.42024975329095854953503450563, 0.73451716972853960228923945920, 3.48848833204778501510986461515, 4.18406685128348719835927162480, 5.27082646789614655187763602393, 5.81582463711673513488573734027, 7.04566694454643836671471550184, 8.231675053085121830784608659456, 8.712878994571449101850647079410, 9.926913618650585394452057255354, 10.99390400388172762923055382722

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