Properties

Label 2-648-72.13-c1-0-38
Degree $2$
Conductor $648$
Sign $-0.926 - 0.376i$
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  + (−0.228 − 1.39i)2-s + (−1.89 + 0.637i)4-s + (0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s + (1.32 + 2.49i)8-s + (0.5 − 1.32i)10-s + (−2.59 + 1.5i)11-s + (−4.58 − 2.64i)13-s + (−1.09 + 0.895i)14-s + (3.18 − 2.41i)16-s − 5.29·17-s − 5.29i·19-s + (−1.96 − 0.395i)20-s + (2.68 + 3.28i)22-s + (2.64 − 4.58i)23-s + ⋯
L(s)  = 1  + (−0.161 − 0.986i)2-s + (−0.947 + 0.318i)4-s + (0.387 + 0.223i)5-s + (−0.188 − 0.327i)7-s + (0.467 + 0.883i)8-s + (0.158 − 0.418i)10-s + (−0.783 + 0.452i)11-s + (−1.27 − 0.733i)13-s + (−0.292 + 0.239i)14-s + (0.796 − 0.604i)16-s − 1.28·17-s − 1.21i·19-s + (−0.438 − 0.0884i)20-s + (0.572 + 0.700i)22-s + (0.551 − 0.955i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.926 - 0.376i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.926 - 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0825043 + 0.422196i\)
\(L(\frac12)\) \(\approx\) \(0.0825043 + 0.422196i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.228 + 1.39i)T \)
3 \( 1 \)
good5 \( 1 + (-0.866 - 0.5i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.58 + 2.64i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + (-2.64 + 4.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.19 - 3i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.29iT - 37T^{2} \)
41 \( 1 + (2.64 - 4.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.16 - 5.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + (3.46 + 2i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 3T + 73T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.06 + 3.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.10168127950713953594247232224, −9.545389811777469913555280164443, −8.496974588926836917395291152963, −7.56827394065689934058481859448, −6.57123357413224580941572521031, −5.07377626130179729602361108569, −4.49584078773634706878379468877, −2.91491504242085576772986753048, −2.24525093534973007623253565392, −0.22913639274563606743476023325, 2.00538733315545875183692457873, 3.71181534504462240383229583980, 5.01339477535728544559620011304, 5.58040050039025390057999249252, 6.66885729277324977141089068187, 7.46145811927063710119110242141, 8.406829010092760242083667346453, 9.264998330707033342571486322129, 9.828638761215641245903146831653, 10.83670594561262912160634306991

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