↑

Properties

Label 2-1008-21.5-c3-0-28
Degree $2$
Conductor $1008$
Sign $-0.0698 + 0.997i$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.36 + 7.56i)5-s + (−14.4 + 11.5i)7-s + (−7.60 + 4.39i)11-s − 11.8i·13-s + (22.2 + 38.6i)17-s + (−10.0 − 5.82i)19-s + (−123. − 71.3i)23-s + (24.3 + 42.1i)25-s + 234. i·29-s + (−252. + 145. i)31-s + (−23.9 − 160. i)35-s + (44.4 − 76.9i)37-s + 145.·41-s − 144.·43-s + (120. − 208. i)47-s + ⋯
L(s)  = 1  + (−0.390 + 0.676i)5-s + (−0.782 + 0.622i)7-s + (−0.208 + 0.120i)11-s − 0.252i·13-s + (0.318 + 0.550i)17-s + (−0.121 − 0.0703i)19-s + (−1.11 − 0.646i)23-s + (0.194 + 0.337i)25-s + 1.49i·29-s + (−1.46 + 0.845i)31-s + (−0.115 − 0.772i)35-s + (0.197 − 0.342i)37-s + 0.555·41-s − 0.512·43-s + (0.372 − 0.646i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0698 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0698 + 0.997i$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -0.0698 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3288723396\)
\(L(\frac12)\) \(\approx\) \(0.3288723396\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (14.4 - 11.5i)T \)
good5 \( 1 + (4.36 - 7.56i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.60 - 4.39i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 11.8iT - 2.19e3T^{2} \)
17 \( 1 + (-22.2 - 38.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (10.0 + 5.82i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (123. + 71.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 234. iT - 2.43e4T^{2} \)
31 \( 1 + (252. - 145. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-44.4 + 76.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 145.T + 6.89e4T^{2} \)
43 \( 1 + 144.T + 7.95e4T^{2} \)
47 \( 1 + (-120. + 208. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-263. + 152. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (3.54 + 6.13i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (149. + 86.4i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (243. + 421. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 653. iT - 3.57e5T^{2} \)
73 \( 1 + (99.0 - 57.1i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-147. + 255. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 877.T + 5.71e5T^{2} \)
89 \( 1 + (-710. + 1.23e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 738. iT - 9.12e5T^{2} \)
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

−9.309700785009032029736775081888, −8.611964330538276535051909725482, −7.60442800604942961508788697067, −6.84026980396380970466815986966, −6.00497316086183740284275191713, −5.12071915815518975560697893802, −3.74193587176613913923615605718, −3.08238671080243526610775382402, −1.90698099821409430155158002506, −0.099513709393977380031489422703, 0.906576007330222998678357179156, 2.42073746560803241414026935293, 3.72907729534579555234280762528, 4.34691147330697481223120906682, 5.54279467244378834382253193178, 6.36517703304372028569361828182, 7.46499759656246092829478338281, 7.992057073721152156152595444469, 9.102974922564112267077732738932, 9.720204698879250526306824842012

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