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Properties

Label 2-648-9.7-c3-0-16
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  + (2 + 3.46i)5-s + (−1.5 + 2.59i)7-s + (14 − 24.2i)11-s + (5.5 + 9.52i)13-s − 44·17-s + 29·19-s + (86 + 148. i)23-s + (54.5 − 94.3i)25-s + (96 − 166. i)29-s + (−58 − 100. i)31-s − 12·35-s − 69·37-s + (192 + 332. i)41-s + (−164 + 284. i)43-s + (78 − 135. i)47-s + ⋯
L(s)  = 1  + (0.178 + 0.309i)5-s + (−0.0809 + 0.140i)7-s + (0.383 − 0.664i)11-s + (0.117 + 0.203i)13-s − 0.627·17-s + 0.350·19-s + (0.779 + 1.35i)23-s + (0.435 − 0.755i)25-s + (0.614 − 1.06i)29-s + (−0.336 − 0.582i)31-s − 0.0579·35-s − 0.306·37-s + (0.731 + 1.26i)41-s + (−0.581 + 1.00i)43-s + (0.242 − 0.419i)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: $\chi_{648} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.081759197\)
\(L(\frac12)\) \(\approx\) \(2.081759197\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-14 + 24.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-5.5 - 9.52i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 44T + 4.91e3T^{2} \)
19 \( 1 - 29T + 6.85e3T^{2} \)
23 \( 1 + (-86 - 148. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-96 + 166. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (58 + 100. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 69T + 5.06e4T^{2} \)
41 \( 1 + (-192 - 332. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (164 - 284. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-78 + 135. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 392T + 1.48e5T^{2} \)
59 \( 1 + (-206 - 356. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-212.5 + 368. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (128.5 + 222. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.00e3T + 3.57e5T^{2} \)
73 \( 1 + 359T + 3.89e5T^{2} \)
79 \( 1 + (438.5 - 759. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (164 - 284. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.57e3T + 7.04e5T^{2} \)
97 \( 1 + (-741.5 + 1.28e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.14216929907730292376989992801, −9.350130940176037922680062527259, −8.544202177901722602809517226757, −7.54691566322269069005738199093, −6.54511044920518010015174810621, −5.82734509053004913085101402410, −4.63965408864599461481208023207, −3.50088807492538801063425082066, −2.41652323201728577711516370380, −0.933253730624204428339741375135, 0.804159124062839267316897135724, 2.13974071610175097490448164495, 3.46768152953684714389152572140, 4.62963590334389746907152016623, 5.43436024597814758713018789838, 6.73589275710730070435016942947, 7.24650867154556919791910586934, 8.678281800984904516333734871005, 9.020534435284298184562811431769, 10.22506019527761033686665983662

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