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Properties

Label 1-547-547.113-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.977 - 0.210i$
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  + (0.0287 + 0.999i)2-s + (−0.222 + 0.974i)3-s + (−0.998 + 0.0575i)4-s + (0.905 − 0.423i)5-s + (−0.980 − 0.194i)6-s + (−0.177 + 0.984i)7-s + (−0.0862 − 0.996i)8-s + (−0.900 − 0.433i)9-s + (0.449 + 0.893i)10-s + (0.428 + 0.903i)11-s + (0.166 − 0.986i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.211 + 0.977i)15-s + (0.993 − 0.114i)16-s + (0.932 + 0.359i)17-s + ⋯
L(s)  = 1  + (0.0287 + 0.999i)2-s + (−0.222 + 0.974i)3-s + (−0.998 + 0.0575i)4-s + (0.905 − 0.423i)5-s + (−0.980 − 0.194i)6-s + (−0.177 + 0.984i)7-s + (−0.0862 − 0.996i)8-s + (−0.900 − 0.433i)9-s + (0.449 + 0.893i)10-s + (0.428 + 0.903i)11-s + (0.166 − 0.986i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.211 + 0.977i)15-s + (0.993 − 0.114i)16-s + (0.932 + 0.359i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.977 - 0.210i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1297689568 + 1.217394087i\)
\(L(\frac12)\) \(\approx\) \(-0.1297689568 + 1.217394087i\)
\(L(1)\) \(\approx\) \(0.5660627660 + 0.8563518097i\)
\(L(1)\) \(\approx\) \(0.5660627660 + 0.8563518097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.0287 + 0.999i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.905 - 0.423i)T \)
7 \( 1 + (-0.177 + 0.984i)T \)
11 \( 1 + (0.428 + 0.903i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.932 + 0.359i)T \)
19 \( 1 + (-0.944 + 0.327i)T \)
23 \( 1 + (0.838 - 0.544i)T \)
29 \( 1 + (0.962 + 0.272i)T \)
31 \( 1 + (-0.650 + 0.759i)T \)
37 \( 1 + (-0.991 - 0.126i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.857 - 0.514i)T \)
47 \( 1 + (-0.200 + 0.979i)T \)
53 \( 1 + (0.740 - 0.671i)T \)
59 \( 1 + (-0.845 + 0.534i)T \)
61 \( 1 + (-0.459 - 0.888i)T \)
67 \( 1 + (-0.937 + 0.349i)T \)
71 \( 1 + (0.469 - 0.882i)T \)
73 \( 1 + (0.211 - 0.977i)T \)
79 \( 1 + (-0.0172 + 0.999i)T \)
83 \( 1 + (0.987 + 0.160i)T \)
89 \( 1 + (-0.539 - 0.842i)T \)
97 \( 1 + (-0.439 - 0.898i)T \)
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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

−23.02373260477141468931255685795, −22.08188808219844584578814771302, −21.23552905526277253953085178421, −20.37319886147825097334495681831, −19.466923357475548975626161360361, −18.85236058909898952408081303615, −18.047415612049043356274373594534, −17.21024314461558171102498846304, −16.780016829839366608759081584964, −14.86095901645401188316371904761, −13.84652906165496859366672551317, −13.5256169438591415284510330333, −12.7804028444652763052125810616, −11.69418213119919592623549541262, −10.785227658972402623685714711670, −10.29023471712914367213099780440, −9.07515209027452898940840642954, −8.1178885639506021234912100532, −6.99425108541143332587646363977, −5.98922494943556499785609199191, −5.16511610834514925988998245261, −3.54809784717529386756388167211, −2.82623007077971888513313719491, −1.56617603044724087742396136089, −0.70568038187248698309966936834, 1.67286862381095421829139359724, 3.27794520153817397385318644969, 4.53961693341713056010945475254, 5.12958652317875034431211544160, 6.11039595733271300124388084176, 6.70429288169828964868419276981, 8.46604859647323813021033925244, 8.970975254686754147849804912508, 9.70597551680065181231309065084, 10.50536787888785863221243626431, 12.10152875859462075178685910735, 12.673862240691254285449125483677, 13.99226023831068609692145843750, 14.68284804442857494480647673854, 15.32283128204207359006312257164, 16.429465670376885849306781026017, 16.785912327796389251757821611832, 17.67974647113076957890620771538, 18.48214185462496635325662798855, 19.55568727821836669808908490084, 21.04521053114134778555485785042, 21.350408824945479460793240391205, 22.169092824188133365962353868853, 22.99230829691766710906141762864, 23.758720586918052419245328966869

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