↑

Properties

Label 2-1008-21.2-c2-0-9
Degree $2$
Conductor $1008$
Sign $0.397 - 0.917i$
Analytic cond. $27.4660$
Root an. cond. $5.24080$
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  + (−1.22 − 0.707i)5-s + (−6.5 − 2.59i)7-s + (−6.12 + 3.53i)11-s + 15·13-s + (9.79 − 5.65i)17-s + (−6.5 + 11.2i)19-s + (−19.5 − 11.3i)23-s + (−11.5 − 19.9i)25-s + 22.6i·29-s + (1.5 + 2.59i)31-s + (6.12 + 7.77i)35-s + (−8.5 + 14.7i)37-s + 80.6i·41-s + 85·43-s + (62.4 + 36.0i)47-s + ⋯
L(s)  = 1  + (−0.244 − 0.141i)5-s + (−0.928 − 0.371i)7-s + (−0.556 + 0.321i)11-s + 1.15·13-s + (0.576 − 0.332i)17-s + (−0.342 + 0.592i)19-s + (−0.851 − 0.491i)23-s + (−0.460 − 0.796i)25-s + 0.780i·29-s + (0.0483 + 0.0838i)31-s + (0.174 + 0.222i)35-s + (−0.229 + 0.397i)37-s + 1.96i·41-s + 1.97·43-s + (1.32 + 0.767i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.397 - 0.917i$
Analytic conductor: \(27.4660\)
Root analytic conductor: \(5.24080\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1),\ 0.397 - 0.917i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.155262140\)
\(L(\frac12)\) \(\approx\) \(1.155262140\)
\(L(2)\) 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 \( 1 \)
7 \( 1 + (6.5 + 2.59i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.12 - 3.53i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 15T + 169T^{2} \)
17 \( 1 + (-9.79 + 5.65i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.5 - 11.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (19.5 + 11.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 22.6iT - 841T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (8.5 - 14.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 80.6iT - 1.68e3T^{2} \)
43 \( 1 - 85T + 1.84e3T^{2} \)
47 \( 1 + (-62.4 - 36.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (29.3 - 16.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (78.3 - 45.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-36 + 62.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.5 - 37.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 52.3iT - 5.04e3T^{2} \)
73 \( 1 + (-47.5 - 82.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-34.5 + 59.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 60.8iT - 6.88e3T^{2} \)
89 \( 1 + (117. + 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 16T + 9.40e3T^{2} \)
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

−10.02375259549510589645288052330, −9.134353172095253107002242307145, −8.183782006092250870508331979216, −7.51992563098499170617814567732, −6.40016633293207713285343821173, −5.83897559115442374706069078168, −4.49937841827387046102345269952, −3.68947910265967532936563385404, −2.63184626224489193992625927549, −1.03706032957268335909723535898, 0.43011152148520765474455225986, 2.15029142113237372708498678718, 3.35692932381284964919021160274, 4.03489177231261257232370131593, 5.61388162320183071262403076263, 5.99940538329425796228326822905, 7.12528391079973924076113859918, 7.953924447935939730324666054526, 8.859527320998204889089982261850, 9.534050045901934844436037900632

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