Properties

Label 2-672-168.53-c0-0-0
Degree $2$
Conductor $672$
Sign $0.832 + 0.553i$
Analytic cond. $0.335371$
Root an. cond. $0.579112$
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)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 0.999·15-s + (0.499 + 0.866i)21-s + 0.999·27-s + 29-s + (−0.5 + 0.866i)31-s + (0.499 + 0.866i)33-s − 0.999·35-s + (−0.499 + 0.866i)45-s + (−0.499 − 0.866i)49-s + (−0.5 + 0.866i)53-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 0.999·15-s + (0.499 + 0.866i)21-s + 0.999·27-s + 29-s + (−0.5 + 0.866i)31-s + (0.499 + 0.866i)33-s − 0.999·35-s + (−0.499 + 0.866i)45-s + (−0.499 − 0.866i)49-s + (−0.5 + 0.866i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(0.335371\)
Root analytic conductor: \(0.579112\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7581926398\)
\(L(\frac12)\) \(\approx\) \(0.7581926398\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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

−10.70073126480046450237190066788, −9.883583031263375174942489316694, −8.802826805526790771060949719812, −8.328915675856932775456466862810, −7.08589775662895294288499797197, −6.00106889817306809408986297823, −4.92983018325448709752031177170, −4.28444397760405380763367379383, −3.35025979848401591248573946837, −0.997713551704961223929152415114, 1.80434783672156620923385269613, 2.90533663940179130952613923919, 4.43812409956674834889354206479, 5.53234022831618700437095351706, 6.48932399175406306709175052839, 7.22706844495125827639929926091, 7.982592131622134851783817364615, 8.933107251918218814393240957626, 10.08841910434541675615845718302, 11.07881001987800070929954829119

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