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Properties

Label 2-648-9.4-c3-0-25
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

\(L(2)\) \(\approx\) \(1.635629423\)
\(L(\frac12)\) \(\approx\) \(1.635629423\)
\(L(\frac{5}{2})\) 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

−9.705442943488627302536992430161, −9.517377452981288729040101594810, −8.099824969658361213689297638015, −7.15517390852697276661644232836, −6.64726285703222029146915871993, −5.32388264196708886643441223009, −4.26190156154995160052998218185, −3.42584620501000330047134401024, −1.86568405869479346196455974915, −0.52905647721533198470473183562, 1.15308100433438079451612575080, 2.80415422233293758430000314879, 3.43347335047732781255646721776, 5.04853774915963060023383305051, 5.88129886873754225330631503353, 6.61395897162721266958476941329, 7.79432931021386428096195479283, 8.777908651249621726125500917504, 9.353711418763977708710569056692, 10.30740204895596792771543032545

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