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Properties

Label 1-648-648.563-r0-0-0
Degree $1$
Conductor $648$
Sign $0.700 - 0.713i$
Analytic cond. $3.00929$
Root an. cond. $3.00929$
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  + (0.597 − 0.802i)5-s + (0.835 + 0.549i)7-s + (0.993 − 0.116i)11-s + (0.286 − 0.957i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (−0.835 + 0.549i)23-s + (−0.286 − 0.957i)25-s + (−0.686 + 0.727i)29-s + (0.0581 + 0.998i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (−0.973 + 0.230i)41-s + (0.396 − 0.918i)43-s + (−0.0581 + 0.998i)47-s + ⋯
L(s)  = 1  + (0.597 − 0.802i)5-s + (0.835 + 0.549i)7-s + (0.993 − 0.116i)11-s + (0.286 − 0.957i)13-s + (−0.173 − 0.984i)17-s + (0.173 − 0.984i)19-s + (−0.835 + 0.549i)23-s + (−0.286 − 0.957i)25-s + (−0.686 + 0.727i)29-s + (0.0581 + 0.998i)31-s + (0.939 − 0.342i)35-s + (0.939 + 0.342i)37-s + (−0.973 + 0.230i)41-s + (0.396 − 0.918i)43-s + (−0.0581 + 0.998i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(3.00929\)
Root analytic conductor: \(3.00929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (0:\ ),\ 0.700 - 0.713i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.672212493 - 0.7021703569i\)
\(L(\frac12)\) \(\approx\) \(1.672212493 - 0.7021703569i\)
\(L(1)\) \(\approx\) \(1.314664153 - 0.2541259615i\)
\(L(1)\) \(\approx\) \(1.314664153 - 0.2541259615i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (0.835 + 0.549i)T \)
11 \( 1 + (0.993 - 0.116i)T \)
13 \( 1 + (0.286 - 0.957i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.835 + 0.549i)T \)
29 \( 1 + (-0.686 + 0.727i)T \)
31 \( 1 + (0.0581 + 0.998i)T \)
37 \( 1 + (0.939 + 0.342i)T \)
41 \( 1 + (-0.973 + 0.230i)T \)
43 \( 1 + (0.396 - 0.918i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.993 + 0.116i)T \)
61 \( 1 + (-0.893 + 0.448i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (-0.973 - 0.230i)T \)
83 \( 1 + (-0.973 - 0.230i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.597 + 0.802i)T \)
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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

−22.892181962812718597313708524093, −22.11947056748127907656617065529, −21.38304232943636676470955308956, −20.62985711688239001884778197767, −19.69085957014618846818546685767, −18.72308695007778059375649685186, −18.11761208559813911061531579960, −17.0432681219018097060871267788, −16.75829114885269255256575612532, −15.26669804484528464800699469374, −14.42313309577075715891322452665, −14.09850751920114911665902274082, −13.07995352707060373650995662207, −11.79992568258571616318300833749, −11.20750247957608240147695625718, −10.26061130231449987239408463142, −9.51007830376815944860160990118, −8.376825555499102719241256985458, −7.45317027758710602279271583176, −6.44603288278444217798346083111, −5.8360186043997695985089528650, −4.29838787221105025675667264653, −3.76201983227764653803609669465, −2.146287117637036202331860544028, −1.49349680820089873830811799212, 1.00479280962772847294787575385, 1.959791770099809365215490046985, 3.18806306532381724915595126468, 4.56466982776016129037375565618, 5.27033052863932777696823110941, 6.103215003970009260458432365951, 7.31688916803627556410841232379, 8.42389939652426055415523380996, 9.0365731814705699508150584877, 9.85517324721844405915698255604, 11.10025997827316174056020642350, 11.820281223198331884390740493792, 12.68816257659623225823364424575, 13.63890400836334514114864762520, 14.31776778609317335116716649927, 15.35918989237000575431145868780, 16.13163971344945797264112704982, 17.142060209976198590814619412333, 17.810974199750723666671879139561, 18.38125992143772010808317399963, 19.805057015439231874469788995367, 20.25110666601086409457636604011, 21.14478941554406170180984518913, 21.933222118712624608407159191287, 22.53863194073110941216417851171

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