Properties

Label 2-648-648.131-c1-0-91
Degree $2$
Conductor $648$
Sign $0.678 + 0.734i$
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  + (1.35 + 0.391i)2-s + (0.952 − 1.44i)3-s + (1.69 + 1.06i)4-s + (0.133 − 2.28i)5-s + (1.86 − 1.59i)6-s + (0.141 + 0.598i)7-s + (1.88 + 2.10i)8-s + (−1.18 − 2.75i)9-s + (1.07 − 3.05i)10-s + (−2.30 − 3.51i)11-s + (3.15 − 1.43i)12-s + (0.356 + 3.05i)13-s + (−0.0415 + 0.868i)14-s + (−3.18 − 2.37i)15-s + (1.73 + 3.60i)16-s + (3.30 − 3.93i)17-s + ⋯
L(s)  = 1  + (0.960 + 0.276i)2-s + (0.550 − 0.835i)3-s + (0.846 + 0.532i)4-s + (0.0595 − 1.02i)5-s + (0.759 − 0.650i)6-s + (0.0535 + 0.226i)7-s + (0.666 + 0.745i)8-s + (−0.394 − 0.918i)9-s + (0.340 − 0.966i)10-s + (−0.696 − 1.05i)11-s + (0.910 − 0.414i)12-s + (0.0989 + 0.846i)13-s + (−0.0110 + 0.232i)14-s + (−0.821 − 0.612i)15-s + (0.433 + 0.900i)16-s + (0.801 − 0.954i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88266 - 1.26218i\)
\(L(\frac12)\) \(\approx\) \(2.88266 - 1.26218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.391i)T \)
3 \( 1 + (-0.952 + 1.44i)T \)
good5 \( 1 + (-0.133 + 2.28i)T + (-4.96 - 0.580i)T^{2} \)
7 \( 1 + (-0.141 - 0.598i)T + (-6.25 + 3.14i)T^{2} \)
11 \( 1 + (2.30 + 3.51i)T + (-4.35 + 10.1i)T^{2} \)
13 \( 1 + (-0.356 - 3.05i)T + (-12.6 + 2.99i)T^{2} \)
17 \( 1 + (-3.30 + 3.93i)T + (-2.95 - 16.7i)T^{2} \)
19 \( 1 + (3.42 - 2.87i)T + (3.29 - 18.7i)T^{2} \)
23 \( 1 + (-7.04 - 1.67i)T + (20.5 + 10.3i)T^{2} \)
29 \( 1 + (4.65 - 6.25i)T + (-8.31 - 27.7i)T^{2} \)
31 \( 1 + (3.35 - 1.00i)T + (25.9 - 17.0i)T^{2} \)
37 \( 1 + (7.56 + 1.33i)T + (34.7 + 12.6i)T^{2} \)
41 \( 1 + (-4.42 - 1.90i)T + (28.1 + 29.8i)T^{2} \)
43 \( 1 + (-6.11 - 3.06i)T + (25.6 + 34.4i)T^{2} \)
47 \( 1 + (-1.04 + 3.50i)T + (-39.2 - 25.8i)T^{2} \)
53 \( 1 + (-1.52 + 2.63i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.96 - 12.1i)T + (-23.3 - 54.1i)T^{2} \)
61 \( 1 + (-3.52 + 3.32i)T + (3.54 - 60.8i)T^{2} \)
67 \( 1 + (0.454 + 0.610i)T + (-19.2 + 64.1i)T^{2} \)
71 \( 1 + (5.34 - 1.94i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-6.64 - 2.41i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (12.3 - 5.33i)T + (54.2 - 57.4i)T^{2} \)
83 \( 1 + (-6.05 + 2.60i)T + (56.9 - 60.3i)T^{2} \)
89 \( 1 + (1.90 - 5.22i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.401 + 6.89i)T + (-96.3 + 11.2i)T^{2} \)
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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

−10.76037821181750949135006815779, −9.076462387483574733456176621528, −8.682847855774776625278856655207, −7.65033205721975147901274038793, −6.92619541520188337114716970939, −5.72964398258354169014781996635, −5.14948841918338967549462274403, −3.75886832463645096513576192959, −2.74751516605242693466899536397, −1.38033941402491720897848832775, 2.23342766520337215795550304033, 3.05608669505102581015461137411, 4.01261674637346158334581700426, 4.99082274626847330304379924182, 5.92538407569313426498833935261, 7.15615282376157863263706781633, 7.78734375699677331646500606002, 9.178184911141857201390900895968, 10.35125965595798861319027790694, 10.51504792164574077296175043394

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