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Properties

Label 2-648-648.403-c0-0-0
Degree $2$
Conductor $648$
Sign $0.952 + 0.305i$
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  + (−0.0581 + 0.998i)2-s + (−0.286 − 0.957i)3-s + (−0.993 − 0.116i)4-s + (0.973 − 0.230i)6-s + (0.173 − 0.984i)8-s + (−0.835 + 0.549i)9-s + (1.36 − 1.44i)11-s + (0.173 + 0.984i)12-s + (0.973 + 0.230i)16-s + (−1.67 − 0.611i)17-s + (−0.500 − 0.866i)18-s + (1.28 − 0.469i)19-s + (1.36 + 1.44i)22-s + (−0.993 + 0.116i)24-s + (0.893 − 0.448i)25-s + ⋯
L(s)  = 1  + (−0.0581 + 0.998i)2-s + (−0.286 − 0.957i)3-s + (−0.993 − 0.116i)4-s + (0.973 − 0.230i)6-s + (0.173 − 0.984i)8-s + (−0.835 + 0.549i)9-s + (1.36 − 1.44i)11-s + (0.173 + 0.984i)12-s + (0.973 + 0.230i)16-s + (−1.67 − 0.611i)17-s + (−0.500 − 0.866i)18-s + (1.28 − 0.469i)19-s + (1.36 + 1.44i)22-s + (−0.993 + 0.116i)24-s + (0.893 − 0.448i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7465588105\)
\(L(\frac12)\) \(\approx\) \(0.7465588105\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0581 - 0.998i)T \)
3 \( 1 + (0.286 + 0.957i)T \)
good5 \( 1 + (-0.893 + 0.448i)T^{2} \)
7 \( 1 + (0.286 - 0.957i)T^{2} \)
11 \( 1 + (-1.36 + 1.44i)T + (-0.0581 - 0.998i)T^{2} \)
13 \( 1 + (-0.597 - 0.802i)T^{2} \)
17 \( 1 + (1.67 + 0.611i)T + (0.766 + 0.642i)T^{2} \)
19 \( 1 + (-1.28 + 0.469i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.286 + 0.957i)T^{2} \)
29 \( 1 + (-0.396 - 0.918i)T^{2} \)
31 \( 1 + (0.686 - 0.727i)T^{2} \)
37 \( 1 + (-0.173 + 0.984i)T^{2} \)
41 \( 1 + (-0.0333 - 0.572i)T + (-0.993 + 0.116i)T^{2} \)
43 \( 1 + (-0.0333 - 0.111i)T + (-0.835 + 0.549i)T^{2} \)
47 \( 1 + (0.686 + 0.727i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.33 + 1.41i)T + (-0.0581 + 0.998i)T^{2} \)
61 \( 1 + (-0.973 + 0.230i)T^{2} \)
67 \( 1 + (0.997 - 0.656i)T + (0.396 - 0.918i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.290 - 1.64i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.993 + 0.116i)T^{2} \)
83 \( 1 + (0.0890 - 1.52i)T + (-0.993 - 0.116i)T^{2} \)
89 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.770 - 0.182i)T + (0.893 + 0.448i)T^{2} \)
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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

−10.97603328086643660869951693997, −9.395108970375940938009238841400, −8.828691545542036510734425793953, −8.029422269089734275647499848343, −6.91070035813930646370828715688, −6.50853258100468184879538747736, −5.58969101765563584309242088252, −4.50581731844867470998973319414, −3.06216764030538813238612896493, −1.02034719433780609902134972522, 1.76495102094618918388633879513, 3.28826893397058446410372558137, 4.27259003590593880044747512308, 4.84555867092153676812339207519, 6.14078482307713607365130002833, 7.33294650972278274714713025902, 8.859143780540661612363085724214, 9.173589406055665491812771386563, 10.05958402821631067233828465407, 10.72656898499300663679348440968

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