↑

Properties

Label 2-1008-63.59-c1-0-15
Degree $2$
Conductor $1008$
Sign $0.988 - 0.149i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.859 − 1.50i)3-s − 4.19·5-s + (−1.67 + 2.05i)7-s + (−1.52 − 2.58i)9-s + 1.66i·11-s + (5.64 + 3.25i)13-s + (−3.60 + 6.31i)15-s + (1.45 − 2.52i)17-s + (2.39 − 1.38i)19-s + (1.64 + 4.27i)21-s + 2.21i·23-s + 12.6·25-s + (−5.19 + 0.0719i)27-s + (5.69 − 3.28i)29-s + (−0.414 + 0.239i)31-s + ⋯
L(s)  = 1  + (0.495 − 0.868i)3-s − 1.87·5-s + (−0.631 + 0.775i)7-s + (−0.507 − 0.861i)9-s + 0.503i·11-s + (1.56 + 0.903i)13-s + (−0.931 + 1.63i)15-s + (0.353 − 0.611i)17-s + (0.549 − 0.317i)19-s + (0.359 + 0.933i)21-s + 0.462i·23-s + 2.52·25-s + (−0.999 + 0.0138i)27-s + (1.05 − 0.610i)29-s + (−0.0743 + 0.0429i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.988 - 0.149i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.988 - 0.149i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.186533063\)
\(L(\frac12)\) \(\approx\) \(1.186533063\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.859 + 1.50i)T \)
7 \( 1 + (1.67 - 2.05i)T \)
good5 \( 1 + 4.19T + 5T^{2} \)
11 \( 1 - 1.66iT - 11T^{2} \)
13 \( 1 + (-5.64 - 3.25i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.45 + 2.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.39 + 1.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.21iT - 23T^{2} \)
29 \( 1 + (-5.69 + 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.414 - 0.239i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.378 + 0.655i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.769 - 1.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.79 - 8.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.05 - 7.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.11 - 4.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.426 + 0.739i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.89 - 2.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.69 - 13.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.89iT - 71T^{2} \)
73 \( 1 + (6.22 + 3.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.52 - 9.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.162 + 0.280i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.86 + 4.95i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.22 + 2.44i)T + (48.5 - 84.0i)T^{2} \)
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

−9.737346357880541999264467133758, −8.843380930188493007821299221872, −8.348611398006663022981091503506, −7.47108423718835740231779631253, −6.84382723704366042126424102215, −5.94695237051766672339574263273, −4.46158634254992583897761299175, −3.54465709183102220985949955037, −2.76000823104019512958048750492, −1.04182673628219053375406826765, 0.68211364636371083336686748378, 3.27161139252071141448473189069, 3.53008201049222277229701543719, 4.27668602325735980308394188501, 5.47415218432291316209858627330, 6.70466224289311418564732617783, 7.72630270972489846219101280205, 8.335392116713995313494798642224, 8.829636523133824598169019348170, 10.23457454443023668646050986430

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