↑

Properties

Label 2-1008-252.103-c1-0-5
Degree $2$
Conductor $1008$
Sign $-0.906 + 0.421i$
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  + 1.73i·3-s + (−0.621 + 0.358i)5-s + (1 + 2.44i)7-s − 2.99·9-s + (−1.5 − 0.866i)11-s + (−3.62 − 2.09i)13-s + (−0.621 − 1.07i)15-s + (−2.74 + 1.58i)17-s + (0.5 − 0.866i)19-s + (−4.24 + 1.73i)21-s + (−2.37 + 1.37i)23-s + (−2.24 + 3.88i)25-s − 5.19i·27-s + (0.621 + 1.07i)29-s + 4·31-s + ⋯
L(s)  = 1  + 0.999i·3-s + (−0.277 + 0.160i)5-s + (0.377 + 0.925i)7-s − 0.999·9-s + (−0.452 − 0.261i)11-s + (−1.00 − 0.579i)13-s + (−0.160 − 0.277i)15-s + (−0.665 + 0.384i)17-s + (0.114 − 0.198i)19-s + (−0.925 + 0.377i)21-s + (−0.495 + 0.286i)23-s + (−0.448 + 0.776i)25-s − 0.999i·27-s + (0.115 + 0.199i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.906 + 0.421i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5003296656\)
\(L(\frac12)\) \(\approx\) \(0.5003296656\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 + (0.621 - 0.358i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.62 + 2.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.74 - 1.58i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.37 - 1.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.621 - 1.07i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-1.62 + 2.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.74 + 5.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.74 + 3.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 + (0.621 + 1.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 2.02iT - 79T^{2} \)
83 \( 1 + (5.74 + 9.94i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (14.2 + 8.21i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.74 - 3.31i)T + (48.5 - 84.0i)T^{2} \)
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

−10.32361542661153647577319832597, −9.712123462590023090839124449357, −8.747987864470179945576308858250, −8.200274269001825862998491632874, −7.17131595802915403622706426324, −5.83065426427342253174640711386, −5.27067791058013345960568064351, −4.35015608404402612386948109657, −3.20704602361657472259761900935, −2.28568788920532865355485580165, 0.21356037335337680504613656016, 1.72417244252240958177411378734, 2.81403401935160102730646761174, 4.29009301114648518056278975440, 5.01210310203040208151908902213, 6.36800151107488288288808299437, 6.99334210566847781730953686519, 7.86425036624057919168965651325, 8.285923729011171776321887924326, 9.543872367672269742080675774890

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