Properties

Label 2-648-648.547-c0-0-0
Degree $2$
Conductor $648$
Sign $0.925 - 0.378i$
Analytic cond. $0.323394$
Root an. cond. $0.568677$
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  + (−0.993 + 0.116i)2-s + (−0.835 − 0.549i)3-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)6-s + (−0.939 + 0.342i)8-s + (0.396 + 0.918i)9-s + (−0.113 + 1.94i)11-s + (−0.939 − 0.342i)12-s + (0.893 − 0.448i)16-s + (0.914 − 0.767i)17-s + (−0.499 − 0.866i)18-s + (−0.0890 − 0.0747i)19-s + (−0.113 − 1.94i)22-s + (0.973 + 0.230i)24-s + (0.597 + 0.802i)25-s + ⋯
L(s)  = 1  + (−0.993 + 0.116i)2-s + (−0.835 − 0.549i)3-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)6-s + (−0.939 + 0.342i)8-s + (0.396 + 0.918i)9-s + (−0.113 + 1.94i)11-s + (−0.939 − 0.342i)12-s + (0.893 − 0.448i)16-s + (0.914 − 0.767i)17-s + (−0.499 − 0.866i)18-s + (−0.0890 − 0.0747i)19-s + (−0.113 − 1.94i)22-s + (0.973 + 0.230i)24-s + (0.597 + 0.802i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 0.925 - 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4706234896\)
\(L(\frac12)\) \(\approx\) \(0.4706234896\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.993 - 0.116i)T \)
3 \( 1 + (0.835 + 0.549i)T \)
good5 \( 1 + (-0.597 - 0.802i)T^{2} \)
7 \( 1 + (0.835 - 0.549i)T^{2} \)
11 \( 1 + (0.113 - 1.94i)T + (-0.993 - 0.116i)T^{2} \)
13 \( 1 + (0.286 + 0.957i)T^{2} \)
17 \( 1 + (-0.914 + 0.767i)T + (0.173 - 0.984i)T^{2} \)
19 \( 1 + (0.0890 + 0.0747i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.835 + 0.549i)T^{2} \)
29 \( 1 + (0.686 + 0.727i)T^{2} \)
31 \( 1 + (0.0581 - 0.998i)T^{2} \)
37 \( 1 + (0.939 - 0.342i)T^{2} \)
41 \( 1 + (-1.65 - 0.193i)T + (0.973 + 0.230i)T^{2} \)
43 \( 1 + (-1.65 - 1.09i)T + (0.396 + 0.918i)T^{2} \)
47 \( 1 + (0.0581 + 0.998i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.103 + 1.78i)T + (-0.993 + 0.116i)T^{2} \)
61 \( 1 + (-0.893 - 0.448i)T^{2} \)
67 \( 1 + (0.227 + 0.526i)T + (-0.686 + 0.727i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (0.744 - 0.270i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (-0.973 + 0.230i)T^{2} \)
83 \( 1 + (0.344 - 0.0403i)T + (0.973 - 0.230i)T^{2} \)
89 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (1.22 - 0.615i)T + (0.597 - 0.802i)T^{2} \)
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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

−10.82369901267038326559391169907, −9.832105055862695734345685834786, −9.369919716939497033302790145269, −7.86695247471374961981471133562, −7.40042217735921427792719822435, −6.65320774968517640783442776619, −5.58961662306339961281334963569, −4.61806776610519818156107521118, −2.61078885212782939094053098120, −1.38273988979150669222123159700, 0.936710690892133042356387586057, 2.95710725679239105414924224411, 4.01120808046171849116260450403, 5.73513663005178467150210492265, 6.03026691148224118689099022854, 7.25950900294398972108283494182, 8.358137111283800554311494557515, 8.973216857984328672005046551052, 10.02642587484754597601955668526, 10.71596210082002642398925309192

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