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Properties

Label 2-648-9.4-c3-0-21
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  + (5.70 − 9.88i)5-s + (14.9 + 25.8i)7-s + (−33.1 − 57.3i)11-s + (−19.9 + 34.4i)13-s + 107.·17-s + 70.3·19-s + (−3.45 + 5.99i)23-s + (−2.66 − 4.61i)25-s + (−18.3 − 31.7i)29-s + (115. − 200. i)31-s + 340.·35-s + 36.8·37-s + (−214. + 371. i)41-s + (37.1 + 64.3i)43-s + (−26.2 − 45.5i)47-s + ⋯
L(s)  = 1  + (0.510 − 0.884i)5-s + (0.805 + 1.39i)7-s + (−0.907 − 1.57i)11-s + (−0.424 + 0.735i)13-s + 1.53·17-s + 0.849·19-s + (−0.0313 + 0.0543i)23-s + (−0.0213 − 0.0369i)25-s + (−0.117 − 0.203i)29-s + (0.670 − 1.16i)31-s + 1.64·35-s + 0.163·37-s + (−0.817 + 1.41i)41-s + (0.131 + 0.228i)43-s + (−0.0815 − 0.141i)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

\(L(2)\) \(\approx\) \(2.437397561\)
\(L(\frac12)\) \(\approx\) \(2.437397561\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.70 + 9.88i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-14.9 - 25.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (33.1 + 57.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (19.9 - 34.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 107.T + 4.91e3T^{2} \)
19 \( 1 - 70.3T + 6.85e3T^{2} \)
23 \( 1 + (3.45 - 5.99i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (18.3 + 31.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-115. + 200. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 36.8T + 5.06e4T^{2} \)
41 \( 1 + (214. - 371. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-37.1 - 64.3i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (26.2 + 45.5i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 288.T + 1.48e5T^{2} \)
59 \( 1 + (-391. + 678. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (219. + 380. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-109. + 189. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 790.T + 3.57e5T^{2} \)
73 \( 1 - 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + (219. + 380. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-25.4 - 44.0i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 719.T + 7.04e5T^{2} \)
97 \( 1 + (-599. - 1.03e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.844862836053870774517699746931, −9.256880861472392411869199212446, −8.262929323814184828929056846272, −7.921083581727673329895313098580, −6.18891970547830072934430100841, −5.38468503701235760481462720872, −5.01095868415555860512140546012, −3.27311616708989915692539026008, −2.14474416918060711779626530175, −0.893005918541095236221591117660, 1.02569938515430774526045928938, 2.34842932744882721930734083325, 3.49907683495338281909128341131, 4.78144034109563403202799788473, 5.49724037985137457570934949406, 7.08061732797020980394908811383, 7.32966936740005056693696897375, 8.172249047738657004093302237832, 9.803221000780486614011988502846, 10.31378526052484551401613349482

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