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Properties

Label 2-648-216.11-c1-0-25
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: $\chi_{648} (35, \cdot )$
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

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

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