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Properties

Label 2-648-216.13-c1-0-25
Degree $2$
Conductor $648$
Sign $0.999 + 0.00981i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.377i)2-s + (1.71 + 1.02i)4-s + (2.15 − 2.57i)5-s + (−1.78 − 0.649i)7-s + (1.94 + 2.04i)8-s + (3.91 − 2.69i)10-s + (3.21 + 3.83i)11-s + (−3.61 − 0.637i)13-s + (−2.18 − 1.55i)14-s + (1.88 + 3.52i)16-s + (1.74 − 3.02i)17-s + (2.73 − 1.57i)19-s + (6.35 − 2.19i)20-s + (2.94 + 6.44i)22-s + (2.93 − 1.06i)23-s + ⋯
L(s)  = 1  + (0.963 + 0.266i)2-s + (0.857 + 0.514i)4-s + (0.965 − 1.15i)5-s + (−0.674 − 0.245i)7-s + (0.689 + 0.724i)8-s + (1.23 − 0.851i)10-s + (0.970 + 1.15i)11-s + (−1.00 − 0.176i)13-s + (−0.584 − 0.416i)14-s + (0.471 + 0.882i)16-s + (0.422 − 0.732i)17-s + (0.626 − 0.361i)19-s + (1.42 − 0.490i)20-s + (0.626 + 1.37i)22-s + (0.611 − 0.222i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.999 + 0.00981i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.999 + 0.00981i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.03015 - 0.0148698i\)
\(L(\frac12)\) \(\approx\) \(3.03015 - 0.0148698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.377i)T \)
3 \( 1 \)
good5 \( 1 + (-2.15 + 2.57i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.78 + 0.649i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-3.21 - 3.83i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (3.61 + 0.637i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.74 + 3.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.73 + 1.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.93 + 1.06i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (6.61 - 1.16i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.842 + 0.306i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (6.21 + 3.58i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.341 + 1.93i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.36 - 6.39i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.63 + 0.959i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 9.52iT - 53T^{2} \)
59 \( 1 + (6.13 - 7.31i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (2.92 - 8.03i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (13.7 + 2.42i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (1.50 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.472 - 0.818i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.803 + 4.55i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.12 + 0.375i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-7.83 - 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.68 + 1.41i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.50394523184878665936572475079, −9.479004838979899307714560885755, −9.144596055073827658544179316923, −7.52700084381442232742610775550, −6.95856539246314313281354232255, −5.82781042921969773933422240002, −5.04398796144976952145661631493, −4.28916522070866665893931188711, −2.87045067369387005692364679306, −1.55771129734672821395027285280, 1.78082639589017203786972828412, 3.02713056770934395256001585847, 3.61436610242098076622264170262, 5.24800412381192930906471095307, 6.10730081201166686508212563778, 6.57938873285844340756609110313, 7.55910867063462620583360691247, 9.197032132026975858640321937073, 9.884262682030239722639760112625, 10.64116700036075356772293942433

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