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Properties

Label 2-648-9.4-c3-0-29
Degree $2$
Conductor $648$
Sign $-0.342 + 0.939i$
Analytic cond. $38.2332$
Root an. cond. $6.18330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.99 − 15.5i)5-s + (−3.64 − 6.30i)7-s + (−2.30 − 3.99i)11-s + (−14.9 + 25.8i)13-s + 67.9·17-s + 111.·19-s + (109. − 189. i)23-s + (−99.2 − 171. i)25-s + (−17.1 − 29.6i)29-s + (38.8 − 67.2i)31-s − 130.·35-s − 347.·37-s + (−117. + 203. i)41-s + (−26.6 − 46.2i)43-s + (192. + 333. i)47-s + ⋯
L(s)  = 1  + (0.804 − 1.39i)5-s + (−0.196 − 0.340i)7-s + (−0.0632 − 0.109i)11-s + (−0.318 + 0.551i)13-s + 0.969·17-s + 1.34·19-s + (0.990 − 1.71i)23-s + (−0.793 − 1.37i)25-s + (−0.109 − 0.190i)29-s + (0.225 − 0.389i)31-s − 0.632·35-s − 1.54·37-s + (−0.446 + 0.773i)41-s + (−0.0946 − 0.163i)43-s + (0.598 + 1.03i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ -0.342 + 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.121214752\)
\(L(\frac12)\) \(\approx\) \(2.121214752\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-8.99 + 15.5i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (3.64 + 6.30i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (2.30 + 3.99i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (14.9 - 25.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 67.9T + 4.91e3T^{2} \)
19 \( 1 - 111.T + 6.85e3T^{2} \)
23 \( 1 + (-109. + 189. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (17.1 + 29.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-38.8 + 67.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 347.T + 5.06e4T^{2} \)
41 \( 1 + (117. - 203. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (26.6 + 46.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-192. - 333. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 461.T + 1.48e5T^{2} \)
59 \( 1 + (3.58 - 6.20i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (208. + 360. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-434. + 752. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 585.T + 3.57e5T^{2} \)
73 \( 1 + 733.T + 3.89e5T^{2} \)
79 \( 1 + (585. + 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-33.7 - 58.4i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 965.T + 7.04e5T^{2} \)
97 \( 1 + (0.716 + 1.24i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.742868409457927588556113389127, −9.127765653645314834345752603337, −8.304310661031059123840961596473, −7.25956808634925590203357726464, −6.16759250105873987566083242151, −5.19526337362196610396857461801, −4.55641428642779360088943321205, −3.11165186844255843489050075325, −1.63755309745304010662441703927, −0.62574526275503215346349848536, 1.47748251573506453092903554806, 2.87643531316625550844748700859, 3.39127329760242773002813399780, 5.33311041262461161364989911821, 5.75613409084623182201080299785, 7.09604342221339696038062811055, 7.39980501285887293901219075765, 8.840138955338912706258705713339, 9.851133778555146695039282740186, 10.19599171954611281534582382737

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