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Properties

Label 2-648-216.13-c1-0-8
Degree $2$
Conductor $648$
Sign $0.999 - 0.0247i$
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  + (0.691 − 1.23i)2-s + (−1.04 − 1.70i)4-s + (−1.90 + 2.27i)5-s + (1.28 + 0.466i)7-s + (−2.82 + 0.109i)8-s + (1.48 + 3.92i)10-s + (2.78 + 3.31i)11-s + (4.65 + 0.820i)13-s + (1.46 − 1.25i)14-s + (−1.81 + 3.56i)16-s + (−0.356 + 0.618i)17-s + (−6.43 + 3.71i)19-s + (5.87 + 0.880i)20-s + (6.01 − 1.14i)22-s + (4.17 − 1.51i)23-s + ⋯
L(s)  = 1  + (0.488 − 0.872i)2-s + (−0.522 − 0.852i)4-s + (−0.853 + 1.01i)5-s + (0.484 + 0.176i)7-s + (−0.999 + 0.0386i)8-s + (0.470 + 1.24i)10-s + (0.839 + 1.00i)11-s + (1.29 + 0.227i)13-s + (0.390 − 0.336i)14-s + (−0.454 + 0.890i)16-s + (−0.0865 + 0.149i)17-s + (−1.47 + 0.852i)19-s + (1.31 + 0.196i)20-s + (1.28 − 0.243i)22-s + (0.870 − 0.316i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.999 - 0.0247i$
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: \((2,\ 648,\ (\ :1/2),\ 0.999 - 0.0247i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61807 + 0.0200164i\)
\(L(\frac12)\) \(\approx\) \(1.61807 + 0.0200164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.691 + 1.23i)T \)
3 \( 1 \)
good5 \( 1 + (1.90 - 2.27i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-1.28 - 0.466i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.78 - 3.31i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-4.65 - 0.820i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.356 - 0.618i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.43 - 3.71i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.17 + 1.51i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-6.69 + 1.18i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-5.02 + 1.82i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.547 - 0.315i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.356 - 2.02i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-2.43 - 2.90i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (5.10 + 1.85i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (-0.803 + 0.957i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.216 + 0.595i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (14.0 + 2.47i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-0.414 + 0.717i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.29 - 12.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.22 + 6.96i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.29 - 0.227i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (2.66 + 4.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.52 - 1.27i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.77380311193866106476382348470, −10.04981745579818191985746716981, −8.870664903232397176231454301564, −8.104928501576462425831264700523, −6.73294291412959690302093792702, −6.17435685989544019845308331543, −4.54581272949044251759311592378, −4.00100150165212418705757977141, −2.87539562747119401355569635542, −1.54978759367347334054826751450, 0.856006333185496986956836212997, 3.28434031751668822257977492750, 4.25627454863485728526839057789, 4.90453959209708339302649626091, 6.11881850406647442210244197714, 6.86897565749941445710848506883, 8.226713534202580183148096754850, 8.470099671827648090772590209360, 9.134626908392666777176686446290, 10.84236312917138829593796087185

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