Properties

Label 2-648-648.155-c1-0-33
Degree $2$
Conductor $648$
Sign $0.614 + 0.788i$
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  + (−0.847 − 1.13i)2-s + (−0.462 − 1.66i)3-s + (−0.562 + 1.91i)4-s + (0.539 − 0.354i)5-s + (−1.49 + 1.93i)6-s + (0.568 + 0.536i)7-s + (2.64 − 0.989i)8-s + (−2.57 + 1.54i)9-s + (−0.858 − 0.309i)10-s + (2.27 + 4.53i)11-s + (3.46 + 0.0527i)12-s + (2.07 − 0.893i)13-s + (0.125 − 1.09i)14-s + (−0.841 − 0.736i)15-s + (−3.36 − 2.16i)16-s + (3.54 + 4.22i)17-s + ⋯
L(s)  = 1  + (−0.599 − 0.800i)2-s + (−0.266 − 0.963i)3-s + (−0.281 + 0.959i)4-s + (0.241 − 0.158i)5-s + (−0.611 + 0.791i)6-s + (0.214 + 0.202i)7-s + (0.936 − 0.349i)8-s + (−0.857 + 0.514i)9-s + (−0.271 − 0.0979i)10-s + (0.686 + 1.36i)11-s + (0.999 + 0.0152i)12-s + (0.574 − 0.247i)13-s + (0.0334 − 0.293i)14-s + (−0.217 − 0.190i)15-s + (−0.841 − 0.540i)16-s + (0.860 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.614 + 0.788i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.614 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.966048 - 0.471968i\)
\(L(\frac12)\) \(\approx\) \(0.966048 - 0.471968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.847 + 1.13i)T \)
3 \( 1 + (0.462 + 1.66i)T \)
good5 \( 1 + (-0.539 + 0.354i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (-0.568 - 0.536i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (-2.27 - 4.53i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (-2.07 + 0.893i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (-3.54 - 4.22i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (0.124 + 0.104i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-1.52 - 1.61i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (-8.67 + 1.01i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (1.62 + 6.87i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (3.35 - 0.592i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (-2.81 - 2.09i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (0.158 - 2.72i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (10.9 + 2.59i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (-4.44 - 7.69i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.04 + 8.06i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (-14.6 + 4.39i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (9.65 + 1.12i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (10.2 + 3.73i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.89 - 0.688i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (0.407 - 0.303i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (-0.385 + 0.287i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (3.27 + 9.01i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-8.36 - 5.50i)T + (38.4 + 89.0i)T^{2} \)
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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

−10.39968203669553392111397772365, −9.671954600939725208491639358551, −8.660457934068759415544794052275, −7.929064323178586088923680043150, −7.09907822395471880626387172131, −6.07623716314259298248657713231, −4.83339612150074546718166127902, −3.51953299116209813236499216980, −2.08483696190375434441675126190, −1.25820250116494721680675383358, 0.914430389794917158655228585427, 3.14536876402565167247826453423, 4.39619256028789431196521170583, 5.40589133004793694046345240936, 6.17258363135541423645382634109, 7.01037945393460823833391890852, 8.480982072151956675775244663347, 8.734472447447499365483759839782, 9.833408144746463851369949898985, 10.44336518503372293887712464516

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