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Properties

Label 1-644-644.27-r0-0-0
Degree $1$
Conductor $644$
Sign $0.117 - 0.993i$
Analytic cond. $2.99072$
Root an. cond. $2.99072$
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  + (0.841 − 0.540i)3-s + (−0.415 − 0.909i)5-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.959 + 0.281i)13-s + (−0.841 − 0.540i)15-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.654 + 0.755i)25-s + (−0.142 − 0.989i)27-s + (−0.142 + 0.989i)29-s + (0.841 + 0.540i)31-s + (0.959 + 0.281i)33-s + (0.415 − 0.909i)37-s + (0.959 − 0.281i)39-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)3-s + (−0.415 − 0.909i)5-s + (0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.959 + 0.281i)13-s + (−0.841 − 0.540i)15-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.654 + 0.755i)25-s + (−0.142 − 0.989i)27-s + (−0.142 + 0.989i)29-s + (0.841 + 0.540i)31-s + (0.959 + 0.281i)33-s + (0.415 − 0.909i)37-s + (0.959 − 0.281i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.117 - 0.993i$
Analytic conductor: \(2.99072\)
Root analytic conductor: \(2.99072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 644,\ (0:\ ),\ 0.117 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.401050909 - 1.245091892i\)
\(L(\frac12)\) \(\approx\) \(1.401050909 - 1.245091892i\)
\(L(1)\) \(\approx\) \(1.290226832 - 0.5361986738i\)
\(L(1)\) \(\approx\) \(1.290226832 - 0.5361986738i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (0.415 - 0.909i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−22.95765886193668468467487530321, −22.153578037879972268298549509774, −21.46204747045087519253895190005, −20.634084164577274396232414215757, −19.75742964336890112552706435344, −18.93159252345266464307503173663, −18.574826620258615889795721452647, −17.15767186258754601374680818507, −16.33335076620452241680818582295, −15.343956009819503307841467608216, −14.91126421787772054281456012750, −13.95209855109238324298366602513, −13.38059458022624580073283552542, −12.02138895494951652776456813627, −11.06379139089181175369828829227, −10.38842901339784355951384116610, −9.52548993125269580342408290728, −8.21068958968317695651702600543, −8.08016492324519152792508371087, −6.59110143197269700421749028433, −5.84715933595036170833895364235, −4.2199189815259073075672643433, −3.63847017112676364316945206643, −2.827290858131972318046351584794, −1.5185939156940720045355552947, 0.93315669242840711486893636737, 1.86871824971319341480413241427, 3.16412636909413884474296087588, 4.13746523265934303065880116672, 5.02235703024932750522836338570, 6.478772942280490852082590827008, 7.24557761520597087802790097646, 8.18360394080233077393543845410, 9.07542337410041097610745674338, 9.450035741021171826510584227105, 10.998327311482785168299206560902, 12.03170230754428383436363953594, 12.596073632386358749491695266380, 13.55365789367111054047914767975, 14.17231673788076220344530547420, 15.3175799234402075516743955407, 15.87821121961296360267484446660, 16.94562012319895607371714961387, 17.86924988000308157718506376312, 18.661952038910640967850541252921, 19.63937550411739115257400880000, 20.161621904869068252506776158380, 20.78396388316751939534398110688, 21.689524904877104364601287332285, 23.00847338000124756390457474921

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