Properties

Label 1-547-547.10-r0-0-0
Degree $1$
Conductor $547$
Sign $0.709 + 0.704i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (−0.479 + 0.877i)2-s + (−0.900 + 0.433i)3-s + (−0.539 − 0.842i)4-s + (0.997 + 0.0689i)5-s + (0.0517 − 0.998i)6-s + (0.962 − 0.272i)7-s + (0.997 − 0.0689i)8-s + (0.623 − 0.781i)9-s + (−0.539 + 0.842i)10-s + (0.885 + 0.464i)11-s + (0.851 + 0.524i)12-s + (−0.222 + 0.974i)13-s + (−0.222 + 0.974i)14-s + (−0.928 + 0.370i)15-s + (−0.418 + 0.908i)16-s + (0.188 − 0.982i)17-s + ⋯
L(s)  = 1  + (−0.479 + 0.877i)2-s + (−0.900 + 0.433i)3-s + (−0.539 − 0.842i)4-s + (0.997 + 0.0689i)5-s + (0.0517 − 0.998i)6-s + (0.962 − 0.272i)7-s + (0.997 − 0.0689i)8-s + (0.623 − 0.781i)9-s + (−0.539 + 0.842i)10-s + (0.885 + 0.464i)11-s + (0.851 + 0.524i)12-s + (−0.222 + 0.974i)13-s + (−0.222 + 0.974i)14-s + (−0.928 + 0.370i)15-s + (−0.418 + 0.908i)16-s + (0.188 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.709 + 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014814603 + 0.4185057698i\)
\(L(\frac12)\) \(\approx\) \(1.014814603 + 0.4185057698i\)
\(L(1)\) \(\approx\) \(0.8161225552 + 0.3277907988i\)
\(L(1)\) \(\approx\) \(0.8161225552 + 0.3277907988i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.479 + 0.877i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.997 + 0.0689i)T \)
7 \( 1 + (0.962 - 0.272i)T \)
11 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.188 - 0.982i)T \)
19 \( 1 + (0.449 - 0.893i)T \)
23 \( 1 + (-0.832 - 0.553i)T \)
29 \( 1 + (0.509 - 0.860i)T \)
31 \( 1 + (0.770 - 0.636i)T \)
37 \( 1 + (-0.952 - 0.305i)T \)
41 \( 1 + T \)
43 \( 1 + (0.322 - 0.946i)T \)
47 \( 1 + (-0.354 - 0.935i)T \)
53 \( 1 + (-0.928 + 0.370i)T \)
59 \( 1 + (-0.970 + 0.239i)T \)
61 \( 1 + (-0.985 + 0.171i)T \)
67 \( 1 + (0.851 + 0.524i)T \)
71 \( 1 + (-0.792 + 0.609i)T \)
73 \( 1 + (-0.928 - 0.370i)T \)
79 \( 1 + (0.322 - 0.946i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + (-0.985 - 0.171i)T \)
97 \( 1 + (0.256 + 0.966i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.00321187270825661618990161486, −22.182704091826961325119339046757, −21.61099768287939907215258461817, −20.9472397656680239179343163924, −19.8540765489571372592744913251, −18.97586576824626160121508453835, −18.00768471500894985996928818740, −17.60791139653035631389149599372, −17.03973821694675546456972433761, −16.04253446817223891343928756359, −14.397875993242182680705813066770, −13.76969759115579118742876169126, −12.57014694738394865742485544734, −12.184061043297752205625542232398, −11.097735351033547667025607757493, −10.469968346180516386705692494497, −9.62180224819354932088319305794, −8.425138673200575814404541182350, −7.71619162943938670754237605153, −6.296572921941185757521354089747, −5.47681639819954536112940280358, −4.50980905622544660509243026920, −3.0809028204537469516173256839, −1.6318221181901202985647450013, −1.30787037050610229927119671408, 0.94279103088244758530154996088, 2.03743366030405692908941790740, 4.32521709987990650502817877061, 4.82054977569100323865092217281, 5.83928820208563820740448372598, 6.67754123600973351309843825314, 7.38646862001210818000509317899, 8.88495632594599640192895904057, 9.55487065725820945834632169553, 10.274240119644524255690485929025, 11.292939436355006305238086936716, 12.11066737497392349352036912702, 13.72402342821383154754424931581, 14.16342631317116054130095829058, 15.10568412309623457269360088117, 16.0951645453536763313434906668, 16.94304944942267921170830872029, 17.48081433188365663187198466379, 17.99322538648625065712975389577, 18.90405441416404459591687242347, 20.24179933264035693877633918071, 21.1110277500180119879646631521, 22.04855332245267454836895845938, 22.65254779610336969610918762905, 23.601345572642907579215459939613

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