↑

Properties

Label 2-1008-112.51-c0-0-1
Degree $2$
Conductor $1008$
Sign $0.997 + 0.0674i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.258 − 0.965i)5-s + (−0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)10-s + (0.258 − 0.965i)11-s + (1 + i)13-s + (0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (0.707 + 0.707i)20-s + 22-s + (0.707 − 1.22i)23-s + (−0.707 + 1.22i)26-s + (0.866 + 0.5i)28-s + (−0.707 − 0.707i)29-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.258 − 0.965i)5-s + (−0.5 − 0.866i)7-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)10-s + (0.258 − 0.965i)11-s + (1 + i)13-s + (0.707 − 0.707i)14-s + (0.500 − 0.866i)16-s + (0.707 + 0.707i)20-s + 22-s + (0.707 − 1.22i)23-s + (−0.707 + 1.22i)26-s + (0.866 + 0.5i)28-s + (−0.707 − 0.707i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.997 + 0.0674i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.997 + 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9498316252\)
\(L(\frac12)\) \(\approx\) \(0.9498316252\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
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

−9.875904473255781215201332386259, −9.013158482765203736858540063163, −8.486173433749881895052357199133, −7.70004316649583712807313050952, −6.51701473250701652869459753821, −6.19279785774573361250190923312, −4.81347010661278597834797307657, −4.20282713880007494650671736155, −3.28696684098421995474206715054, −0.916555695665794486249253442370, 1.73385282925096996140256921178, 3.07913044444080022296992620655, 3.44869801082743206655978634160, 4.86771540135358274983423582009, 5.77723220967464839080916167677, 6.66070107109294961532046501872, 7.73815043337038344902473299348, 8.818481862990443199099290504530, 9.501406707472411817228088035197, 10.31805453203817913721404139963

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