↑

Properties

Label 2-648-9.2-c2-0-10
Degree $2$
Conductor $648$
Sign $0.939 - 0.342i$
Analytic cond. $17.6567$
Root an. cond. $4.20199$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 + 1.41i)5-s + (1.5 − 2.59i)7-s + (17.1 + 9.89i)11-s + (−3.5 − 6.06i)13-s − 14.1i·17-s − 19·19-s + (−2.44 + 1.41i)23-s + (−8.50 + 14.7i)25-s + (44.0 + 25.4i)29-s + (5 + 8.66i)31-s + 8.48i·35-s + 63·37-s + (44.0 − 25.4i)41-s + (25 − 43.3i)43-s + (36.7 + 21.2i)47-s + ⋯
L(s)  = 1  + (−0.489 + 0.282i)5-s + (0.214 − 0.371i)7-s + (1.55 + 0.899i)11-s + (−0.269 − 0.466i)13-s − 0.831i·17-s − 19-s + (−0.106 + 0.0614i)23-s + (−0.340 + 0.588i)25-s + (1.52 + 0.877i)29-s + (0.161 + 0.279i)31-s + 0.242i·35-s + 1.70·37-s + (1.07 − 0.620i)41-s + (0.581 − 1.00i)43-s + (0.781 + 0.451i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(17.6567\)
Root analytic conductor: \(4.20199\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.777334776\)
\(L(\frac12)\) \(\approx\) \(1.777334776\)
\(L(2)\) 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 \( 1 + (2.44 - 1.41i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-17.1 - 9.89i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.5 + 6.06i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 14.1iT - 289T^{2} \)
19 \( 1 + 19T + 361T^{2} \)
23 \( 1 + (2.44 - 1.41i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-44.0 - 25.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 63T + 1.36e3T^{2} \)
41 \( 1 + (-44.0 + 25.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-25 + 43.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-36.7 - 21.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 73.5iT - 2.80e3T^{2} \)
59 \( 1 + (-85.7 + 49.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (39.5 - 68.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (38.5 + 66.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 79.1iT - 5.04e3T^{2} \)
73 \( 1 + 17T + 5.32e3T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-34.2 - 19.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 42.4iT - 7.92e3T^{2} \)
97 \( 1 + (-48.5 + 84.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.46428851744529425376404532624, −9.497630715973052544454843831582, −8.755799575998596989676031044867, −7.54735018751429241578265043701, −7.03555840131383225820419604932, −6.01296688035664149847174783529, −4.60355872574662389896616011506, −3.97301554363362746631238963667, −2.59809377083241434429203207382, −1.04722363394917587348122482322, 0.870306946084058631517255545813, 2.37858769347778194357885008348, 3.92365290992001381831183919651, 4.46904876366091035647250011598, 6.06047954547589638769314100909, 6.47387527085755567522738955974, 7.919504534329600008378555639318, 8.535150941796877128399564492322, 9.289643515136592913957218610605, 10.30732501622625267188212371688

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