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Properties

Label 2-648-9.4-c3-0-8
Degree $2$
Conductor $648$
Sign $-0.939 - 0.342i$
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  + (−9.08 + 15.7i)5-s + (0.411 + 0.712i)7-s + (32.7 + 56.7i)11-s + (−15.9 + 27.5i)13-s + 124.·17-s + 27.8·19-s + (21.5 − 37.3i)23-s + (−102. − 177. i)25-s + (−19.9 − 34.4i)29-s + (−147. + 255. i)31-s − 14.9·35-s − 104.·37-s + (−153. + 266. i)41-s + (180. + 312. i)43-s + (−198. − 344. i)47-s + ⋯
L(s)  = 1  + (−0.812 + 1.40i)5-s + (0.0222 + 0.0384i)7-s + (0.898 + 1.55i)11-s + (−0.339 + 0.587i)13-s + 1.77·17-s + 0.336·19-s + (0.195 − 0.338i)23-s + (−0.821 − 1.42i)25-s + (−0.127 − 0.220i)29-s + (−0.854 + 1.47i)31-s − 0.0722·35-s − 0.462·37-s + (−0.585 + 1.01i)41-s + (0.640 + 1.10i)43-s + (−0.617 − 1.06i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 648,\ (\ :3/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.379923460\)
\(L(\frac12)\) \(\approx\) \(1.379923460\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.08 - 15.7i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-0.411 - 0.712i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-32.7 - 56.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (15.9 - 27.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 124.T + 4.91e3T^{2} \)
19 \( 1 - 27.8T + 6.85e3T^{2} \)
23 \( 1 + (-21.5 + 37.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (19.9 + 34.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (147. - 255. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 104.T + 5.06e4T^{2} \)
41 \( 1 + (153. - 266. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-180. - 312. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (198. + 344. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 107.T + 1.48e5T^{2} \)
59 \( 1 + (94.4 - 163. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-49.5 - 85.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (212. - 368. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 445.T + 3.57e5T^{2} \)
73 \( 1 + 499.T + 3.89e5T^{2} \)
79 \( 1 + (285. + 493. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (654. + 1.13e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + (472. + 818. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.33413478571631785135283595580, −9.966929616774174901421752228909, −8.837520053457757097287001682790, −7.47757213934273766017662799180, −7.23770839781293046169612922084, −6.33031513981930267417000824037, −4.92506044821481245064456174046, −3.84905901404761364164469600272, −2.99979474802813970241092116733, −1.60962688880860810806467503402, 0.44765400869901656363301805326, 1.24831796389745582490219618672, 3.29852017618117473289401513329, 4.00865596334124051192012362529, 5.30925894305062834114222246171, 5.82883420692286476517917298729, 7.42422173343910784535650545228, 8.056512181064439179085184743951, 8.873005279105180215585068595377, 9.525835108507050014553333527649

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