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Properties

Label 2-648-216.11-c1-0-8
Degree $2$
Conductor $648$
Sign $0.960 - 0.279i$
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  + (−1.12 − 0.856i)2-s + (0.532 + 1.92i)4-s + (0.588 + 0.214i)5-s + (4.57 + 0.806i)7-s + (1.05 − 2.62i)8-s + (−0.479 − 0.745i)10-s + (1.37 + 3.76i)11-s + (0.583 + 0.695i)13-s + (−4.45 − 4.82i)14-s + (−3.43 + 2.05i)16-s + (−2.52 + 1.45i)17-s + (−2.50 + 4.34i)19-s + (−0.0993 + 1.24i)20-s + (1.68 − 5.41i)22-s + (−0.819 − 4.64i)23-s + ⋯
L(s)  = 1  + (−0.795 − 0.605i)2-s + (0.266 + 0.963i)4-s + (0.263 + 0.0958i)5-s + (1.72 + 0.304i)7-s + (0.371 − 0.928i)8-s + (−0.151 − 0.235i)10-s + (0.413 + 1.13i)11-s + (0.161 + 0.192i)13-s + (−1.19 − 1.28i)14-s + (−0.857 + 0.513i)16-s + (−0.613 + 0.353i)17-s + (−0.574 + 0.995i)19-s + (−0.0222 + 0.279i)20-s + (0.358 − 1.15i)22-s + (−0.170 − 0.968i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.960 - 0.279i$
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: \((2,\ 648,\ (\ :1/2),\ 0.960 - 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24060 + 0.176686i\)
\(L(\frac12)\) \(\approx\) \(1.24060 + 0.176686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.856i)T \)
3 \( 1 \)
good5 \( 1 + (-0.588 - 0.214i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-4.57 - 0.806i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-1.37 - 3.76i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-0.583 - 0.695i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.52 - 1.45i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.50 - 4.34i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.819 + 4.64i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.21 - 5.21i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.29 - 0.756i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (3.72 - 2.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.41 - 6.44i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-5.48 + 1.99i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.38 + 7.85i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 3.06T + 53T^{2} \)
59 \( 1 + (-0.537 + 1.47i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-3.32 - 0.586i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-9.02 + 7.57i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (3.19 + 5.53i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.18 - 3.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.52 + 6.58i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (6.42 - 7.65i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-1.88 - 1.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.312 - 0.113i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.60908582102580599433368276232, −9.849579595737942747686605781627, −8.719928941579040818069637554706, −8.285443326129353183488889127857, −7.31950230400915487359600789640, −6.31242322686985830453323656636, −4.80631866788768369563675856585, −4.04397535664807426608263617304, −2.25228887406410718352605419002, −1.60502433424732968756882349196, 0.981016578017521980277465552820, 2.23366151643332560918544776030, 4.21109281504919247946019304276, 5.25870030815280331990549865643, 6.03708875975324500740130081148, 7.20846266629862271457178459679, 7.960536704330433500043544134087, 8.724468987481923462983441093050, 9.362141159403562156837956472382, 10.62993813268180559292524455753

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