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Properties

Label 2-648-9.4-c3-0-3
Degree $2$
Conductor $648$
Sign $-0.173 - 0.984i$
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  + (−1 + 1.73i)5-s + (−12 − 20.7i)7-s + (−22 − 38.1i)11-s + (−11 + 19.0i)13-s − 50·17-s + 44·19-s + (−28 + 48.4i)23-s + (60.5 + 104. i)25-s + (99 + 171. i)29-s + (80 − 138. i)31-s + 48·35-s − 162·37-s + (−99 + 171. i)41-s + (−26 − 45.0i)43-s + (264 + 457. i)47-s + ⋯
L(s)  = 1  + (−0.0894 + 0.154i)5-s + (−0.647 − 1.12i)7-s + (−0.603 − 1.04i)11-s + (−0.234 + 0.406i)13-s − 0.713·17-s + 0.531·19-s + (−0.253 + 0.439i)23-s + (0.483 + 0.838i)25-s + (0.633 + 1.09i)29-s + (0.463 − 0.802i)31-s + 0.231·35-s − 0.719·37-s + (−0.377 + 0.653i)41-s + (−0.0922 − 0.159i)43-s + (0.819 + 1.41i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
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: \((2,\ 648,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6599684407\)
\(L(\frac12)\) \(\approx\) \(0.6599684407\)
\(L(\frac{5}{2})\) 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 \( 1 + (1 - 1.73i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (12 + 20.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (22 + 38.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (11 - 19.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 50T + 4.91e3T^{2} \)
19 \( 1 - 44T + 6.85e3T^{2} \)
23 \( 1 + (28 - 48.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-99 - 171. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-80 + 138. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 162T + 5.06e4T^{2} \)
41 \( 1 + (99 - 171. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (26 + 45.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-264 - 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 242T + 1.48e5T^{2} \)
59 \( 1 + (334 - 578. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (275 + 476. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (94 - 162. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 728T + 3.57e5T^{2} \)
73 \( 1 - 154T + 3.89e5T^{2} \)
79 \( 1 + (-328 - 568. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-118 - 204. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 714T + 7.04e5T^{2} \)
97 \( 1 + (-239 - 413. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.52292302953065660462191370122, −9.578734759383615817765128385501, −8.704723714868392880586206525073, −7.62214447456477314984682258437, −6.93510175111827270636373129756, −5.99655875446824650775123878046, −4.82984354306585173770823619764, −3.71161943398741022857133480950, −2.84189157796045839808486460297, −1.08974842838240832900997237513, 0.20753574627567907397600727409, 2.12923742344217128470869358102, 2.96005027876523609254972131000, 4.44725423681082230657047285755, 5.31629077333186188592788258123, 6.31705470112171227026398270020, 7.21529938937790780939825899586, 8.309988181002088436631245163516, 9.005143251180905756070549648234, 9.980151991246154127191365621933

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