Properties

Label 2-648-216.11-c1-0-10
Degree $2$
Conductor $648$
Sign $-0.183 - 0.983i$
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  + (−0.160 + 1.40i)2-s + (−1.94 − 0.450i)4-s + (1.93 + 0.705i)5-s + (0.744 + 0.131i)7-s + (0.944 − 2.66i)8-s + (−1.30 + 2.60i)10-s + (0.949 + 2.60i)11-s + (−0.421 − 0.502i)13-s + (−0.303 + 1.02i)14-s + (3.59 + 1.75i)16-s + (3.79 − 2.18i)17-s + (0.155 − 0.270i)19-s + (−3.45 − 2.24i)20-s + (−3.81 + 0.916i)22-s + (1.32 + 7.53i)23-s + ⋯
L(s)  = 1  + (−0.113 + 0.993i)2-s + (−0.974 − 0.225i)4-s + (0.866 + 0.315i)5-s + (0.281 + 0.0496i)7-s + (0.333 − 0.942i)8-s + (−0.411 + 0.825i)10-s + (0.286 + 0.786i)11-s + (−0.116 − 0.139i)13-s + (−0.0811 + 0.273i)14-s + (0.898 + 0.438i)16-s + (0.919 − 0.531i)17-s + (0.0357 − 0.0619i)19-s + (−0.773 − 0.502i)20-s + (−0.813 + 0.195i)22-s + (0.276 + 1.57i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.183 - 0.983i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986727 + 1.18799i\)
\(L(\frac12)\) \(\approx\) \(0.986727 + 1.18799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.160 - 1.40i)T \)
3 \( 1 \)
good5 \( 1 + (-1.93 - 0.705i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.744 - 0.131i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.949 - 2.60i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.421 + 0.502i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-3.79 + 2.18i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.155 + 0.270i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 - 7.53i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-5.89 - 4.94i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-4.16 + 0.733i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.70 - 1.56i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.31 + 5.14i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.00 - 1.09i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.89 - 10.7i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 0.876T + 53T^{2} \)
59 \( 1 + (2.94 - 8.09i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-12.1 - 2.14i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-3.32 + 2.79i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (7.08 + 12.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.17 + 12.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.09 - 8.44i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-7.10 + 8.47i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (5.55 + 3.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.59 + 0.579i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.36460590312392865283054032009, −9.815193938961436316075135679758, −9.112440188096697537731186520878, −8.015593017465574322740039917049, −7.21536064191303600792419562758, −6.39579944666513099360518491344, −5.43897015470285460874812673696, −4.70255368745639947364827545589, −3.23581782368230091303614133580, −1.50747203025334349653778685089, 1.02876377829801302881143774356, 2.27004990026780793908449826304, 3.47561590147970395173565146524, 4.66659753849327743948325047933, 5.56843689540554950849204137329, 6.59990880075746955081996133175, 8.254325691156966598264146405862, 8.541701367919022568934514331076, 9.811441552721868187883899714173, 10.13050209049707691264311382136

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