↑

Properties

Label 2-1008-28.23-c0-0-0
Degree $2$
Conductor $1008$
Sign $0.832 - 0.553i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
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  + (−0.5 + 0.866i)7-s + 13-s + (1.5 + 0.866i)19-s + (0.5 + 0.866i)25-s + (−1.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + (−1 + 1.73i)61-s + (1.5 − 0.866i)67-s + (−0.5 − 0.866i)73-s + (−1.5 − 0.866i)79-s + (−0.5 + 0.866i)91-s − 2·97-s + (1.5 + 0.866i)103-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + 13-s + (1.5 + 0.866i)19-s + (0.5 + 0.866i)25-s + (−1.5 + 0.866i)31-s + (0.5 − 0.866i)37-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + (−1 + 1.73i)61-s + (1.5 − 0.866i)67-s + (−0.5 − 0.866i)73-s + (−1.5 − 0.866i)79-s + (−0.5 + 0.866i)91-s − 2·97-s + (1.5 + 0.866i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.014099444\)
\(L(\frac12)\) \(\approx\) \(1.014099444\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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

−10.22996797482655364332287354211, −9.213959693212811832901045368249, −8.838940813472546015952107453341, −7.71941402892878545355882274792, −6.89292234167930793368298031164, −5.77225174810134724728055507973, −5.35643391439854627798698363140, −3.81218958854971985760883422615, −3.05808760714726225457124507477, −1.60865109116096409236982239783, 1.12867824155953033217823816685, 2.87279342299094172998720509360, 3.78503175262904335487523784256, 4.77723423080864028136684722525, 5.90656580584473475784357339622, 6.74660570911995325271643102217, 7.52917761798086390072529399693, 8.399741372738161205283802021324, 9.441866102141382422424133166892, 9.943164268862807875968095886512

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