Properties

Degree $2$
Conductor $1568$
Sign $0.968 + 0.250i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 − 1.07i)3-s + (1.61 + 2.80i)5-s + (0.736 + 1.27i)9-s + (3.23 − 5.60i)11-s − 0.763·13-s + 4·15-s + (2.23 − 3.87i)17-s + (−0.618 − 1.07i)19-s + (−2 − 3.46i)23-s + (−2.73 + 4.73i)25-s + 5.52·27-s − 4.47·29-s + (−1.23 + 2.14i)31-s + (−3.99 − 6.92i)33-s + (2.23 + 3.87i)37-s + ⋯
L(s)  = 1  + (0.356 − 0.618i)3-s + (0.723 + 1.25i)5-s + (0.245 + 0.424i)9-s + (0.975 − 1.68i)11-s − 0.211·13-s + 1.03·15-s + (0.542 − 0.939i)17-s + (−0.141 − 0.245i)19-s + (−0.417 − 0.722i)23-s + (−0.547 + 0.947i)25-s + 1.06·27-s − 0.830·29-s + (−0.222 + 0.384i)31-s + (−0.696 − 1.20i)33-s + (0.367 + 0.636i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
Motivic weight: \(1\)
Character: $\chi_{1568} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.430530251\)
\(L(\frac12)\) \(\approx\) \(2.430530251\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (-0.618 + 1.07i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.61 - 2.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.23 + 5.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.763T + 13T^{2} \)
17 \( 1 + (-2.23 + 3.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.618 + 1.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (1.23 - 2.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.47T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + (-5.23 - 9.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.61 - 7.99i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.61 - 9.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (1.47 - 2.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.47 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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

−9.337668940882816269874739626295, −8.624726065687283472881688273242, −7.61859524478455111339523926514, −7.03212659754555051377460567314, −6.21096437715703021896409158587, −5.63322644018677180133594010184, −4.20086765688749714920559095768, −3.01130244317053824634602620685, −2.47452599232644446512216152154, −1.12119898595768259774953155189, 1.27834936501295305893782705802, 2.14106352375399772421463332286, 3.91358325363214017269895142489, 4.21340540097511975254601781891, 5.27750310080199303799399355696, 6.04699202568584551542536949452, 7.11196009201750804713482240043, 7.981649269363956728136887973432, 9.072059975504621395228337445566, 9.428223225127409840227544167436

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