↑

Properties

Label 2-1008-63.41-c1-0-16
Degree $2$
Conductor $1008$
Sign $0.690 + 0.723i$
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  + (−1.62 + 0.608i)3-s + (−1.94 + 3.36i)5-s + (−2.09 − 1.60i)7-s + (2.26 − 1.97i)9-s + (−3.41 + 1.97i)11-s + (−2.46 − 1.42i)13-s + (1.10 − 6.64i)15-s − 0.742·17-s − 1.78i·19-s + (4.38 + 1.33i)21-s + (5.41 + 3.12i)23-s + (−5.07 − 8.78i)25-s + (−2.46 + 4.57i)27-s + (−2.50 + 1.44i)29-s + (3.04 + 1.75i)31-s + ⋯
L(s)  = 1  + (−0.936 + 0.351i)3-s + (−0.870 + 1.50i)5-s + (−0.793 − 0.608i)7-s + (0.753 − 0.657i)9-s + (−1.03 + 0.594i)11-s + (−0.684 − 0.395i)13-s + (0.285 − 1.71i)15-s − 0.179·17-s − 0.409i·19-s + (0.956 + 0.291i)21-s + (1.12 + 0.651i)23-s + (−1.01 − 1.75i)25-s + (−0.474 + 0.880i)27-s + (−0.464 + 0.268i)29-s + (0.546 + 0.315i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3529532778\)
\(L(\frac12)\) \(\approx\) \(0.3529532778\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 0.608i)T \)
7 \( 1 + (2.09 + 1.60i)T \)
good5 \( 1 + (1.94 - 3.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.41 - 1.97i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.46 + 1.42i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.742T + 17T^{2} \)
19 \( 1 + 1.78iT - 19T^{2} \)
23 \( 1 + (-5.41 - 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.50 - 1.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.04 - 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + (-5.24 + 9.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.471 + 0.816i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.09 - 1.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.0105 + 0.0183i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.13 + 1.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.72 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + 4.85iT - 73T^{2} \)
79 \( 1 + (-1.81 - 3.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.02 + 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.26T + 89T^{2} \)
97 \( 1 + (16.2 - 9.40i)T + (48.5 - 84.0i)T^{2} \)
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

−10.11014187463346469402155204120, −9.402946046952492658968112413565, −7.76574241590161220132992489763, −7.15869296769848901532316513217, −6.71781815341890013378833983179, −5.56363052844689206794988377137, −4.53712591570948275790829015021, −3.54292730013684800995740788553, −2.69410365529096878970950366838, −0.24895938295555901318463973182, 0.877413117282533253946510825222, 2.59064416870987579750919519354, 4.12823436998652760401130055636, 4.99059897892774524140967711084, 5.60627142436739537232202219245, 6.60833549251225364968653440892, 7.66942882061732186804985141988, 8.319961429057584213319183282678, 9.215683385565206223430191500643, 10.02506161058347066261349885968

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