↑

Properties

Label 2-1638-273.62-c1-0-5
Degree $2$
Conductor $1638$
Sign $-0.884 - 0.466i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.583i·5-s + (−1.97 − 1.76i)7-s − 0.999·8-s + (0.504 − 0.291i)10-s + (0.339 + 0.587i)11-s + (1.22 + 3.38i)13-s + (0.539 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (−2.81 + 4.87i)17-s + (0.189 − 0.327i)19-s + (0.504 + 0.291i)20-s + (−0.339 + 0.587i)22-s + (1.66 − 0.963i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.260i·5-s + (−0.745 − 0.666i)7-s − 0.353·8-s + (0.159 − 0.0921i)10-s + (0.102 + 0.177i)11-s + (0.340 + 0.940i)13-s + (0.144 − 0.692i)14-s + (−0.125 − 0.216i)16-s + (−0.683 + 1.18i)17-s + (0.0434 − 0.0752i)19-s + (0.112 + 0.0651i)20-s + (−0.0722 + 0.125i)22-s + (0.347 − 0.200i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.884 - 0.466i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055622511\)
\(L(\frac12)\) \(\approx\) \(1.055622511\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (1.97 + 1.76i)T \)
13 \( 1 + (-1.22 - 3.38i)T \)
good5 \( 1 + 0.583iT - 5T^{2} \)
11 \( 1 + (-0.339 - 0.587i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.81 - 4.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.189 + 0.327i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.66 + 0.963i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.454 + 0.262i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.05T + 31T^{2} \)
37 \( 1 + (7.31 - 4.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.16 - 4.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.30 + 3.98i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 - 12.8iT - 53T^{2} \)
59 \( 1 + (-6.02 - 3.47i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.36 + 5.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.43 - 5.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.33 + 9.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.53T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (13.2 - 7.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.55 - 4.43i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.451569049417394809590494988149, −8.915446873359561131632714486390, −8.105678706645918417331814665591, −7.01320144273587411307076931325, −6.65567712611522405349387286911, −5.78474162880589438203160980566, −4.64440703682637659616651636718, −4.02948497104263352141467314304, −3.07147956526227257020932083410, −1.51511629634681175656246740428, 0.35379952935367595254434576794, 2.03839134238330791039931996327, 3.06924007221083212047313451072, 3.61643512513795363346552745301, 5.04347003315350582045308680045, 5.54812954161537180549213724470, 6.60810318466681920063615662424, 7.25401266480327586461975502488, 8.622057165502543963106418199568, 9.022592072194408273300318450672

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