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Properties

Label 2-648-216.59-c1-0-27
Degree $2$
Conductor $648$
Sign $0.767 - 0.641i$
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  + (1.20 + 0.742i)2-s + (0.898 + 1.78i)4-s + (3.07 − 1.11i)5-s + (1.33 − 0.235i)7-s + (−0.245 + 2.81i)8-s + (4.53 + 0.935i)10-s + (0.982 − 2.69i)11-s + (−2.36 + 2.82i)13-s + (1.78 + 0.709i)14-s + (−2.38 + 3.20i)16-s + (−1.94 − 1.12i)17-s + (−1.22 − 2.12i)19-s + (4.76 + 4.48i)20-s + (3.18 − 2.51i)22-s + (1.08 − 6.17i)23-s + ⋯
L(s)  = 1  + (0.851 + 0.524i)2-s + (0.449 + 0.893i)4-s + (1.37 − 0.500i)5-s + (0.505 − 0.0891i)7-s + (−0.0867 + 0.996i)8-s + (1.43 + 0.295i)10-s + (0.296 − 0.813i)11-s + (−0.656 + 0.782i)13-s + (0.477 + 0.189i)14-s + (−0.596 + 0.802i)16-s + (−0.470 − 0.271i)17-s + (−0.280 − 0.486i)19-s + (1.06 + 1.00i)20-s + (0.679 − 0.537i)22-s + (0.227 − 1.28i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.767 - 0.641i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.767 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.85477 + 1.03595i\)
\(L(\frac12)\) \(\approx\) \(2.85477 + 1.03595i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 - 0.742i)T \)
3 \( 1 \)
good5 \( 1 + (-3.07 + 1.11i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-1.33 + 0.235i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.982 + 2.69i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (2.36 - 2.82i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.94 + 1.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.22 + 2.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.08 + 6.17i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.00 - 4.19i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.65 - 0.820i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (-3.70 - 2.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.28 - 8.67i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.24 + 1.17i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.21 - 12.5i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 4.14T + 53T^{2} \)
59 \( 1 + (3.92 + 10.7i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (-9.59 + 1.69i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.64 + 5.57i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-3.19 + 5.53i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.87 + 4.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.31 - 6.33i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.56 - 1.85i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (1.06 - 0.612i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-16.9 - 6.17i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.87358758942443565067149191211, −9.595776646411458457860434614171, −8.866853376854530300595811934597, −8.013584387668930830015988467143, −6.67451639812122391127434243081, −6.23532635886123098397937194629, −5.02771074006933996881974548878, −4.57219011289188309208722200437, −2.93326164768558004140663182090, −1.78715568436935181157715107928, 1.73243069851121533583687342272, 2.44942058200640081863778327004, 3.81659892346521102572136699597, 5.07411887959761472946625130538, 5.71176231019905021478058779817, 6.63964017862973411073016431059, 7.57084462719961266952711092641, 9.071858305700026360479865078135, 10.07461342948736102564788618655, 10.23477835530272481709941412462

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