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Properties

Label 2-648-9.4-c3-0-27
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  + (7 − 12.1i)5-s + (12 + 20.7i)7-s + (−14 − 24.2i)11-s + (37 − 64.0i)13-s − 82·17-s + 92·19-s + (4 − 6.92i)23-s + (−35.5 − 61.4i)25-s + (−69 − 119. i)29-s + (−40 + 69.2i)31-s + 336·35-s + 30·37-s + (141 − 244. i)41-s + (−2 − 3.46i)43-s + (120 + 207. i)47-s + ⋯
L(s)  = 1  + (0.626 − 1.08i)5-s + (0.647 + 1.12i)7-s + (−0.383 − 0.664i)11-s + (0.789 − 1.36i)13-s − 1.16·17-s + 1.11·19-s + (0.0362 − 0.0628i)23-s + (−0.284 − 0.491i)25-s + (−0.441 − 0.765i)29-s + (−0.231 + 0.401i)31-s + 1.62·35-s + 0.133·37-s + (0.537 − 0.930i)41-s + (−0.00709 − 0.0122i)43-s + (0.372 + 0.645i)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

\(L(2)\) \(\approx\) \(2.242390169\)
\(L(\frac12)\) \(\approx\) \(2.242390169\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7 + 12.1i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-12 - 20.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (14 + 24.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-37 + 64.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 82T + 4.91e3T^{2} \)
19 \( 1 - 92T + 6.85e3T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (69 + 119. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (40 - 69.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 30T + 5.06e4T^{2} \)
41 \( 1 + (-141 + 244. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-120 - 207. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 130T + 1.48e5T^{2} \)
59 \( 1 + (-298 + 516. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-109 - 188. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-218 + 377. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 856T + 3.57e5T^{2} \)
73 \( 1 + 998T + 3.89e5T^{2} \)
79 \( 1 + (-16 - 27.7i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (754 + 1.30e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 246T + 7.04e5T^{2} \)
97 \( 1 + (433 + 749. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.844233880186278535647137565263, −8.755379672257053794771258607995, −8.621405379032568533200192710895, −7.52328164065774003601834915746, −5.81504936933466900162022931619, −5.63568407027808041708922415239, −4.63184982051734542147706694343, −3.09802766053267128982406152202, −1.90189234642562492371828503222, −0.66289578997071255824265973458, 1.37942192977206193334380248314, 2.44480035873132897334945559695, 3.83017465570859733431992307674, 4.69510224292460900333135538593, 6.00217826924416378270353643239, 7.02401065066841290790610169606, 7.32910167601967850554579087301, 8.680146680326473530812861359555, 9.640955368111765398935526997859, 10.42673956495899270700617516249

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