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Properties

Label 2-548-548.339-c0-0-0
Degree $2$
Conductor $548$
Sign $0.112 - 0.993i$
Analytic cond. $0.273487$
Root an. cond. $0.522960$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (1.72 + 0.857i)5-s + (−0.982 − 0.183i)8-s + (−0.0922 − 0.995i)9-s + 1.92i·10-s + (−0.576 − 1.48i)13-s + (−0.273 − 0.961i)16-s + (−1.93 + 0.361i)17-s + (0.850 − 0.526i)18-s + (−1.72 + 0.857i)20-s + (1.62 + 2.15i)25-s + (1.07 − 1.17i)26-s + (−0.694 + 0.197i)29-s + (0.739 − 0.673i)32-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (−0.602 + 0.798i)4-s + (1.72 + 0.857i)5-s + (−0.982 − 0.183i)8-s + (−0.0922 − 0.995i)9-s + 1.92i·10-s + (−0.576 − 1.48i)13-s + (−0.273 − 0.961i)16-s + (−1.93 + 0.361i)17-s + (0.850 − 0.526i)18-s + (−1.72 + 0.857i)20-s + (1.62 + 2.15i)25-s + (1.07 − 1.17i)26-s + (−0.694 + 0.197i)29-s + (0.739 − 0.673i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(548\)    =    \(2^{2} \cdot 137\)
Sign: $0.112 - 0.993i$
Analytic conductor: \(0.273487\)
Root analytic conductor: \(0.522960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{548} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 548,\ (\ :0),\ 0.112 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236603195\)
\(L(\frac12)\) \(\approx\) \(1.236603195\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.445 - 0.895i)T \)
137 \( 1 + (-0.932 - 0.361i)T \)
good3 \( 1 + (0.0922 + 0.995i)T^{2} \)
5 \( 1 + (-1.72 - 0.857i)T + (0.602 + 0.798i)T^{2} \)
7 \( 1 + (-0.739 + 0.673i)T^{2} \)
11 \( 1 + (0.273 + 0.961i)T^{2} \)
13 \( 1 + (0.576 + 1.48i)T + (-0.739 + 0.673i)T^{2} \)
17 \( 1 + (1.93 - 0.361i)T + (0.932 - 0.361i)T^{2} \)
19 \( 1 + (-0.445 + 0.895i)T^{2} \)
23 \( 1 + (-0.850 + 0.526i)T^{2} \)
29 \( 1 + (0.694 - 0.197i)T + (0.850 - 0.526i)T^{2} \)
31 \( 1 + (0.445 + 0.895i)T^{2} \)
37 \( 1 + 0.891T + T^{2} \)
41 \( 1 - 1.05iT - T^{2} \)
43 \( 1 + (0.445 - 0.895i)T^{2} \)
47 \( 1 + (-0.982 + 0.183i)T^{2} \)
53 \( 1 + (-0.193 - 0.312i)T + (-0.445 + 0.895i)T^{2} \)
59 \( 1 + (0.982 - 0.183i)T^{2} \)
61 \( 1 + (-0.0505 + 0.544i)T + (-0.982 - 0.183i)T^{2} \)
67 \( 1 + (0.739 - 0.673i)T^{2} \)
71 \( 1 + (-0.273 + 0.961i)T^{2} \)
73 \( 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2} \)
79 \( 1 + (0.0922 - 0.995i)T^{2} \)
83 \( 1 + (0.932 + 0.361i)T^{2} \)
89 \( 1 + (-1.20 - 0.600i)T + (0.602 + 0.798i)T^{2} \)
97 \( 1 + (0.273 + 0.961i)T^{2} \)
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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

−11.08769947641508868581544037934, −10.16822292295727318295248662480, −9.375836608602680988090275048471, −8.628336749039380728921442021358, −7.26302846639243017408979909840, −6.50620835658916484670870087518, −5.90227156903026423430317706035, −5.00326802113448038289750985296, −3.45635010163842468800119508827, −2.41381057626840496323747368944, 1.91223285534010958588880000541, 2.29784456710422679700487460819, 4.39205598647210941607737796597, 4.98344199225755936957744209627, 5.90280129405145854008961567691, 6.91856353652698558377378895327, 8.816869647418232264350783609037, 9.086864832731498594519666679439, 9.992945430543292243234382536407, 10.77846996664428678377917107837

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