Properties

Degree $2$
Conductor $448$
Sign $-0.895 - 0.444i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−1.5 − 0.866i)5-s + (−2 + 1.73i)7-s + (1 − 1.73i)9-s + (−1.5 + 0.866i)11-s + 1.73i·15-s + (−4.5 + 2.59i)17-s + (−3.5 + 6.06i)19-s + (2.5 + 0.866i)21-s + (−7.5 − 4.33i)23-s + (−1 − 1.73i)25-s − 5·27-s + 6·29-s + (2.5 + 4.33i)31-s + (1.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.670 − 0.387i)5-s + (−0.755 + 0.654i)7-s + (0.333 − 0.577i)9-s + (−0.452 + 0.261i)11-s + 0.447i·15-s + (−1.09 + 0.630i)17-s + (−0.802 + 1.39i)19-s + (0.545 + 0.188i)21-s + (−1.56 − 0.902i)23-s + (−0.200 − 0.346i)25-s − 0.962·27-s + 1.11·29-s + (0.449 + 0.777i)31-s + (0.261 + 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.895 - 0.444i$
Motivic weight: \(1\)
Character: $\chi_{448} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.5 + 4.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 2.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (10.5 + 6.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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

−10.49634388134622438449678127577, −9.786471762417294322560496695438, −8.518696948733025503913910717939, −8.035291983805885900797018664265, −6.50166210010701966268844650300, −6.25344521157149162080823110883, −4.66231159497816068174300906934, −3.64271175052699327714804948434, −2.03280589619130762961777842844, 0, 2.56847658675396888151450917094, 3.93210623899632989845404492968, 4.65965766850904310045235773508, 6.02925312566796525387951620130, 7.10175136180396577909991532483, 7.77707565756563504529259897404, 9.048066061743774172803231474715, 9.986530660529737731576087840904, 10.80360376679959592886162014842

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