↑

Properties

Label 2-252-7.6-c8-0-2
Degree $2$
Conductor $252$
Sign $-0.242 - 0.970i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.05e3i·5-s + (582. + 2.32e3i)7-s + 1.65e4·11-s + 4.12e4i·13-s + 1.18e5i·17-s − 1.99e5i·19-s + 5.84e4·23-s − 7.20e5·25-s − 7.85e5·29-s − 2.50e5i·31-s + (2.45e6 − 6.13e5i)35-s − 2.76e6·37-s − 3.01e6i·41-s − 3.93e6·43-s + 9.05e6i·47-s + ⋯
L(s)  = 1  − 1.68i·5-s + (0.242 + 0.970i)7-s + 1.12·11-s + 1.44i·13-s + 1.42i·17-s − 1.52i·19-s + 0.208·23-s − 1.84·25-s − 1.11·29-s − 0.270i·31-s + (1.63 − 0.408i)35-s − 1.47·37-s − 1.06i·41-s − 1.15·43-s + 1.85i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.242 - 0.970i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.242 - 0.970i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.106753123\)
\(L(\frac12)\) \(\approx\) \(1.106753123\)
\(L(5)\) 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 \( 1 \)
7 \( 1 + (-582. - 2.32e3i)T \)
good5 \( 1 + 1.05e3iT - 3.90e5T^{2} \)
11 \( 1 - 1.65e4T + 2.14e8T^{2} \)
13 \( 1 - 4.12e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.18e5iT - 6.97e9T^{2} \)
19 \( 1 + 1.99e5iT - 1.69e10T^{2} \)
23 \( 1 - 5.84e4T + 7.83e10T^{2} \)
29 \( 1 + 7.85e5T + 5.00e11T^{2} \)
31 \( 1 + 2.50e5iT - 8.52e11T^{2} \)
37 \( 1 + 2.76e6T + 3.51e12T^{2} \)
41 \( 1 + 3.01e6iT - 7.98e12T^{2} \)
43 \( 1 + 3.93e6T + 1.16e13T^{2} \)
47 \( 1 - 9.05e6iT - 2.38e13T^{2} \)
53 \( 1 + 7.21e5T + 6.22e13T^{2} \)
59 \( 1 - 1.39e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.42e7iT - 1.91e14T^{2} \)
67 \( 1 - 2.37e7T + 4.06e14T^{2} \)
71 \( 1 - 9.77e6T + 6.45e14T^{2} \)
73 \( 1 - 1.55e7iT - 8.06e14T^{2} \)
79 \( 1 + 5.74e7T + 1.51e15T^{2} \)
83 \( 1 + 3.30e7iT - 2.25e15T^{2} \)
89 \( 1 - 9.46e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.81e6iT - 7.83e15T^{2} \)
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

−11.17880011511597925298049763616, −9.431020520457814729258120686484, −8.990101150920667257032520386936, −8.393338148201936799190841967582, −6.85360192747667590203000743913, −5.74402066411861823077417270626, −4.75209148503497972159816810097, −3.92064589600069453446178015059, −1.98764427479838984074911140217, −1.27316070573104311789264489113, 0.23355501312277267329648222659, 1.66139711101016137842885020401, 3.16830466536225787175846964893, 3.72710940166797043293864296950, 5.35578577231732025627060904343, 6.61423140912469963837002393548, 7.22906390227171368849799190632, 8.136878085211142810424465268466, 9.751365127121845587534827314633, 10.34318890391661534904215585659

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