Properties

Label 2-648-216.11-c1-0-13
Degree $2$
Conductor $648$
Sign $0.689 + 0.724i$
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  + (0.494 − 1.32i)2-s + (−1.51 − 1.31i)4-s + (1.70 + 0.621i)5-s + (3.32 + 0.585i)7-s + (−2.48 + 1.35i)8-s + (1.66 − 1.95i)10-s + (0.942 + 2.58i)11-s + (3.98 + 4.74i)13-s + (2.41 − 4.10i)14-s + (0.561 + 3.96i)16-s + (−2.90 + 1.67i)17-s + (2.87 − 4.98i)19-s + (−1.76 − 3.17i)20-s + (3.89 + 0.0328i)22-s + (0.396 + 2.25i)23-s + ⋯
L(s)  = 1  + (0.349 − 0.936i)2-s + (−0.755 − 0.655i)4-s + (0.763 + 0.277i)5-s + (1.25 + 0.221i)7-s + (−0.878 + 0.477i)8-s + (0.527 − 0.617i)10-s + (0.284 + 0.780i)11-s + (1.10 + 1.31i)13-s + (0.646 − 1.09i)14-s + (0.140 + 0.990i)16-s + (−0.703 + 0.406i)17-s + (0.660 − 1.14i)19-s + (−0.394 − 0.710i)20-s + (0.830 + 0.00700i)22-s + (0.0827 + 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.689 + 0.724i$
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: \((2,\ 648,\ (\ :1/2),\ 0.689 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96042 - 0.840704i\)
\(L(\frac12)\) \(\approx\) \(1.96042 - 0.840704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.494 + 1.32i)T \)
3 \( 1 \)
good5 \( 1 + (-1.70 - 0.621i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-3.32 - 0.585i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.942 - 2.58i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-3.98 - 4.74i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.90 - 1.67i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.87 + 4.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.396 - 2.25i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (7.06 + 5.92i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.38 - 0.244i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.33 + 2.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.54 + 1.84i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.42 + 1.24i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.530 - 3.00i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 0.184T + 53T^{2} \)
59 \( 1 + (-4.44 + 12.2i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (13.9 + 2.45i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-4.70 + 3.94i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.83 + 3.18i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.136 + 0.236i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.710 + 0.846i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (2.40 - 2.86i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (5.59 + 3.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.5 - 5.29i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.75458037107065379367359012667, −9.395798164364446047359704895054, −9.239462120324811712248863638122, −7.995848329845921383073338007926, −6.66847108505720928614646513370, −5.73309153183989949321001562122, −4.70496253087817872398608841503, −3.92063812715172112064832951667, −2.23764704404866365691023451268, −1.62828354946377926350362160103, 1.30785821610596818285842803052, 3.23442173332648945583090890787, 4.36015066580793960451126042006, 5.63256990149743987484319049722, 5.73812482465654559712845238265, 7.16220075567069671021238046971, 8.100690342773872691795324865511, 8.625459610901626109172630250219, 9.567433360713498819262597461161, 10.73445128785291785032398994717

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