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Properties

Label 2-648-9.7-c3-0-28
Degree $2$
Conductor $648$
Sign $0.342 + 0.939i$
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  + (2.90 + 5.02i)5-s + (13.0 − 22.6i)7-s + (18.6 − 32.2i)11-s + (15.1 + 26.2i)13-s − 48.4·17-s + 88.1·19-s + (−42.2 − 73.2i)23-s + (45.6 − 79.0i)25-s + (−88.3 + 153. i)29-s + (−77.6 − 134. i)31-s + 151.·35-s − 258.·37-s + (−110. − 190. i)41-s + (23.8 − 41.3i)43-s + (−64.7 + 112. i)47-s + ⋯
L(s)  = 1  + (0.259 + 0.449i)5-s + (0.705 − 1.22i)7-s + (0.510 − 0.884i)11-s + (0.323 + 0.560i)13-s − 0.691·17-s + 1.06·19-s + (−0.383 − 0.663i)23-s + (0.365 − 0.632i)25-s + (−0.565 + 0.980i)29-s + (−0.450 − 0.779i)31-s + 0.732·35-s − 1.14·37-s + (−0.419 − 0.726i)41-s + (0.0846 − 0.146i)43-s + (−0.200 + 0.348i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.171597379\)
\(L(\frac12)\) \(\approx\) \(2.171597379\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.90 - 5.02i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-13.0 + 22.6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-18.6 + 32.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-15.1 - 26.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 48.4T + 4.91e3T^{2} \)
19 \( 1 - 88.1T + 6.85e3T^{2} \)
23 \( 1 + (42.2 + 73.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (88.3 - 153. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (77.6 + 134. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 258.T + 5.06e4T^{2} \)
41 \( 1 + (110. + 190. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-23.8 + 41.3i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (64.7 - 112. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 577.T + 1.48e5T^{2} \)
59 \( 1 + (-121. - 210. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-343. + 595. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (189. + 327. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 332.T + 3.57e5T^{2} \)
73 \( 1 - 1.07e3T + 3.89e5T^{2} \)
79 \( 1 + (-630. + 1.09e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (613. - 1.06e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.09e3T + 7.04e5T^{2} \)
97 \( 1 + (-564. + 977. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.14991330705265516200263691190, −9.073491868530724503422038312949, −8.267328554969825515729030591221, −7.18978320694356904689797944424, −6.59881392378285908727042425579, −5.42827914511810924515334931505, −4.26377899711407128224327681691, −3.43361761236127947384021016209, −1.89998902285964885557617514139, −0.65622292977134755030525856861, 1.35101203751793166536653914446, 2.32273647499337524917848447906, 3.75742055927645634311422510995, 5.10647105105307194191598118419, 5.50979951150400342751456800502, 6.77189544569441964055080800036, 7.79819558124641940156596367773, 8.733302960480322455418652229662, 9.306383390224999368680407652527, 10.19941248792741198779439429197

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