↑

Properties

Label 2-1028-1028.227-c0-0-0
Degree $2$
Conductor $1028$
Sign $0.401 + 0.915i$
Analytic cond. $0.513038$
Root an. cond. $0.716267$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + (−1.36 + 0.728i)5-s + i·8-s + (0.831 − 0.555i)9-s + (0.728 + 1.36i)10-s + (1.08 + 0.216i)13-s + 16-s + (1.38 − 1.38i)17-s + (−0.555 − 0.831i)18-s + (1.36 − 0.728i)20-s + (0.772 − 1.15i)25-s + (0.216 − 1.08i)26-s + (−1.17 − 0.785i)29-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (−1.36 + 0.728i)5-s + i·8-s + (0.831 − 0.555i)9-s + (0.728 + 1.36i)10-s + (1.08 + 0.216i)13-s + 16-s + (1.38 − 1.38i)17-s + (−0.555 − 0.831i)18-s + (1.36 − 0.728i)20-s + (0.772 − 1.15i)25-s + (0.216 − 1.08i)26-s + (−1.17 − 0.785i)29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1028\)    =    \(2^{2} \cdot 257\)
Sign: $0.401 + 0.915i$
Analytic conductor: \(0.513038\)
Root analytic conductor: \(0.716267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1028} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1028,\ (\ :0),\ 0.401 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8175358000\)
\(L(\frac12)\) \(\approx\) \(0.8175358000\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
257 \( 1 + T \)
good3 \( 1 + (-0.831 + 0.555i)T^{2} \)
5 \( 1 + (1.36 - 0.728i)T + (0.555 - 0.831i)T^{2} \)
7 \( 1 + (-0.980 - 0.195i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-1.08 - 0.216i)T + (0.923 + 0.382i)T^{2} \)
17 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
19 \( 1 + (0.195 - 0.980i)T^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (1.17 + 0.785i)T + (0.382 + 0.923i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.151 - 1.53i)T + (-0.980 + 0.195i)T^{2} \)
41 \( 1 + (-0.728 - 0.598i)T + (0.195 + 0.980i)T^{2} \)
43 \( 1 + (0.831 + 0.555i)T^{2} \)
47 \( 1 + (0.195 - 0.980i)T^{2} \)
53 \( 1 + (-0.598 + 1.11i)T + (-0.555 - 0.831i)T^{2} \)
59 \( 1 + (-0.923 + 0.382i)T^{2} \)
61 \( 1 + (-1.81 - 0.360i)T + (0.923 + 0.382i)T^{2} \)
67 \( 1 + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.195 + 0.980i)T^{2} \)
73 \( 1 + (1.53 - 0.636i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.831 + 0.555i)T^{2} \)
89 \( 1 + (-0.324 - 1.63i)T + (-0.923 + 0.382i)T^{2} \)
97 \( 1 + (0.273 + 0.512i)T + (-0.555 + 0.831i)T^{2} \)
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

−10.04181246497658773908174031871, −9.435692159973406863177319692189, −8.335368064950826428263128492065, −7.62725983227158252728809438057, −6.80365421604406798434057565746, −5.49937122880090942753858536383, −4.22561947372270591140311127065, −3.69096319399655838935129076157, −2.83317436680373887543639993497, −1.06743200876436915685386767151, 1.21694136403485106242364710213, 3.83600179223693207630622167879, 3.94230292690645847036779590057, 5.22918036526820309405737398946, 5.94445372135810752080415484094, 7.30513246819878462869112034306, 7.68671465577390539244574260674, 8.456352918008466695934031050975, 9.078314210260185909283087183510, 10.26536840025289190845690932224

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