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Properties

Label 2-648-216.11-c1-0-6
Degree $2$
Conductor $648$
Sign $-0.141 - 0.989i$
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  + (−1.16 + 0.795i)2-s + (0.734 − 1.86i)4-s + (3.21 + 1.17i)5-s + (−2.85 − 0.504i)7-s + (0.621 + 2.75i)8-s + (−4.69 + 1.18i)10-s + (0.927 + 2.54i)11-s + (1.60 + 1.91i)13-s + (3.74 − 1.68i)14-s + (−2.92 − 2.73i)16-s + (−4.30 + 2.48i)17-s + (−1.31 + 2.27i)19-s + (4.53 − 5.12i)20-s + (−3.11 − 2.24i)22-s + (−0.157 − 0.895i)23-s + ⋯
L(s)  = 1  + (−0.826 + 0.562i)2-s + (0.367 − 0.930i)4-s + (1.43 + 0.523i)5-s + (−1.08 − 0.190i)7-s + (0.219 + 0.975i)8-s + (−1.48 + 0.376i)10-s + (0.279 + 0.768i)11-s + (0.444 + 0.530i)13-s + (1.00 − 0.450i)14-s + (−0.730 − 0.682i)16-s + (−1.04 + 0.603i)17-s + (−0.301 + 0.522i)19-s + (1.01 − 1.14i)20-s + (−0.663 − 0.477i)22-s + (−0.0329 − 0.186i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.141 - 0.989i$
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: \((2,\ 648,\ (\ :1/2),\ -0.141 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703477 + 0.811553i\)
\(L(\frac12)\) \(\approx\) \(0.703477 + 0.811553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 0.795i)T \)
3 \( 1 \)
good5 \( 1 + (-3.21 - 1.17i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (2.85 + 0.504i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.927 - 2.54i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.60 - 1.91i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (4.30 - 2.48i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.31 - 2.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.157 + 0.895i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-7.00 - 5.87i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.90 + 0.512i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-9.78 + 5.64i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.04 - 3.62i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.12 - 2.23i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.225 + 1.28i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (2.96 - 8.13i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.720 - 0.127i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.00 + 4.19i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-5.18 - 8.97i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.34 - 12.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.64 + 6.73i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (3.15 - 3.75i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-4.90 - 2.82i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.126 - 0.0461i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.42526982220240215736227093361, −9.792105093681719196993862344340, −9.283489129789217667486378130383, −8.324327722861606926008083580578, −6.91118143369314931512978713139, −6.49139407148001779213041137955, −5.89425470979131246814461084467, −4.46315439567728689443071284269, −2.70094911740128755258920633316, −1.60919136071763261443357988796, 0.77524657665061208313369022669, 2.30480947335296086189154938893, 3.19176548916452674601674028938, 4.72019892859579838855688637726, 6.29646561414589502577744209854, 6.39844589087824283530810556569, 8.044326852378471341646011493870, 8.921519750887196418104415736065, 9.470992324440314148653246938361, 10.08920268218956824950945080175

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