Properties

Label 2-648-216.11-c1-0-19
Degree $2$
Conductor $648$
Sign $0.883 + 0.468i$
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  + (0.279 + 1.38i)2-s + (−1.84 + 0.774i)4-s + (−2.25 − 0.821i)5-s + (1.95 + 0.344i)7-s + (−1.58 − 2.34i)8-s + (0.508 − 3.35i)10-s + (−0.553 − 1.52i)11-s + (−3.57 − 4.26i)13-s + (0.0681 + 2.80i)14-s + (2.80 − 2.85i)16-s + (6.40 − 3.69i)17-s + (−1.39 + 2.41i)19-s + (4.79 − 0.233i)20-s + (1.95 − 1.19i)22-s + (−0.465 − 2.64i)23-s + ⋯
L(s)  = 1  + (0.197 + 0.980i)2-s + (−0.921 + 0.387i)4-s + (−1.00 − 0.367i)5-s + (0.738 + 0.130i)7-s + (−0.561 − 0.827i)8-s + (0.160 − 1.06i)10-s + (−0.166 − 0.458i)11-s + (−0.992 − 1.18i)13-s + (0.0182 + 0.749i)14-s + (0.700 − 0.714i)16-s + (1.55 − 0.897i)17-s + (−0.320 + 0.554i)19-s + (1.07 − 0.0521i)20-s + (0.416 − 0.254i)22-s + (−0.0970 − 0.550i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.883 + 0.468i$
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: \((2,\ 648,\ (\ :1/2),\ 0.883 + 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.933347 - 0.232097i\)
\(L(\frac12)\) \(\approx\) \(0.933347 - 0.232097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.279 - 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (2.25 + 0.821i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.95 - 0.344i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (0.553 + 1.52i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (3.57 + 4.26i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-6.40 + 3.69i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.465 + 2.64i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.138 + 0.116i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.76 - 0.311i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-3.55 + 2.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.85 - 3.40i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-11.1 + 4.04i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.08 + 11.8i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + (-1.96 + 5.40i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.60 + 1.34i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.38 + 1.16i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.25 + 2.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.79 - 6.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.77 - 8.06i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (0.718 - 0.855i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (9.87 + 5.70i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.25 - 1.91i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.36478688766611100609928179858, −9.440742013994822741716884388621, −8.244707496634187548721253165831, −7.923668613522279696684055415180, −7.24195817750993797263551818444, −5.76083505497032614700778180384, −5.13147999412350456932836130008, −4.16526436372941597098175179011, −3.02963775611055396536314470104, −0.52187707350079529598312162013, 1.54982392488305691111267717627, 2.88401679536777123415913706418, 4.10137186795255877633992144922, 4.65950976061829456391145656096, 5.91188047973405728872789513581, 7.45272722999729335973775601834, 7.88850367284360590117999680847, 9.133206234818501638066996085639, 9.875028559505879528628910075703, 10.90307173100661432214343498256

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