Properties

Label 2-648-9.4-c3-0-20
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  + (−6.67 + 11.5i)5-s + (7.17 + 12.4i)7-s + (−19.5 − 33.8i)11-s + (38.3 − 66.5i)13-s − 62.4·17-s − 39.7·19-s + (64.2 − 111. i)23-s + (−26.7 − 46.2i)25-s + (−32.4 − 56.2i)29-s + (4.56 − 7.91i)31-s − 191.·35-s + 319.·37-s + (−8.78 + 15.2i)41-s + (225. + 390. i)43-s + (290. + 503. i)47-s + ⋯
L(s)  = 1  + (−0.597 + 1.03i)5-s + (0.387 + 0.671i)7-s + (−0.535 − 0.927i)11-s + (0.819 − 1.41i)13-s − 0.890·17-s − 0.480·19-s + (0.582 − 1.00i)23-s + (−0.213 − 0.370i)25-s + (−0.207 − 0.360i)29-s + (0.0264 − 0.0458i)31-s − 0.926·35-s + 1.41·37-s + (−0.0334 + 0.0579i)41-s + (0.799 + 1.38i)43-s + (0.902 + 1.56i)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

\(L(2)\) \(\approx\) \(1.575485235\)
\(L(\frac12)\) \(\approx\) \(1.575485235\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.67 - 11.5i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-7.17 - 12.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (19.5 + 33.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-38.3 + 66.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 62.4T + 4.91e3T^{2} \)
19 \( 1 + 39.7T + 6.85e3T^{2} \)
23 \( 1 + (-64.2 + 111. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (32.4 + 56.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-4.56 + 7.91i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 319.T + 5.06e4T^{2} \)
41 \( 1 + (8.78 - 15.2i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-225. - 390. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-290. - 503. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 329.T + 1.48e5T^{2} \)
59 \( 1 + (-120. + 209. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (248. + 430. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-289. + 500. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 660.T + 3.57e5T^{2} \)
73 \( 1 - 696.T + 3.89e5T^{2} \)
79 \( 1 + (365. + 632. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (545. + 944. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 317.T + 7.04e5T^{2} \)
97 \( 1 + (-742. - 1.28e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.47176825520068441720946429014, −9.074787372990720362230788355743, −8.245225544143994473022391668124, −7.66025664736748475624700253478, −6.39529178170485757368413336940, −5.76850723464646742918208016649, −4.48377993178721356262317994049, −3.21251882227317435614488628154, −2.53245855874569179348782902801, −0.58082202613745626440706377920, 0.946118877479188781225987376750, 2.12049258378068866453526584073, 4.06976479400176359727968306524, 4.37007991785573504338928551822, 5.49233250671400744812954898541, 6.88832444732869969507061377588, 7.50852177902513863342882901973, 8.641493918465976066654572203276, 9.075396148973062908129299322099, 10.24740728663355213102222850217

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