↑

Properties

Label 2-1008-21.5-c3-0-24
Degree $2$
Conductor $1008$
Sign $0.992 + 0.121i$
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  + (−7.89 + 13.6i)5-s + (−10.6 + 15.1i)7-s + (28.2 − 16.3i)11-s − 54.3i·13-s + (−12.3 − 21.3i)17-s + (−16.2 − 9.41i)19-s + (−46.7 − 26.9i)23-s + (−62.0 − 107. i)25-s − 157. i·29-s + (−41.4 + 23.9i)31-s + (−123. − 264. i)35-s + (−48.1 + 83.4i)37-s + 263.·41-s − 258.·43-s + (−62.5 + 108. i)47-s + ⋯
L(s)  = 1  + (−0.705 + 1.22i)5-s + (−0.573 + 0.819i)7-s + (0.774 − 0.446i)11-s − 1.15i·13-s + (−0.175 − 0.304i)17-s + (−0.196 − 0.113i)19-s + (−0.423 − 0.244i)23-s + (−0.496 − 0.859i)25-s − 1.00i·29-s + (−0.240 + 0.138i)31-s + (−0.596 − 1.27i)35-s + (−0.214 + 0.370i)37-s + 1.00·41-s − 0.918·43-s + (−0.194 + 0.336i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.992 + 0.121i$
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: \((2,\ 1008,\ (\ :3/2),\ 0.992 + 0.121i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.309452060\)
\(L(\frac12)\) \(\approx\) \(1.309452060\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.6 - 15.1i)T \)
good5 \( 1 + (7.89 - 13.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.2 + 16.3i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 54.3iT - 2.19e3T^{2} \)
17 \( 1 + (12.3 + 21.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (16.2 + 9.41i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (46.7 + 26.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 157. iT - 2.43e4T^{2} \)
31 \( 1 + (41.4 - 23.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (48.1 - 83.4i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 263.T + 6.89e4T^{2} \)
43 \( 1 + 258.T + 7.95e4T^{2} \)
47 \( 1 + (62.5 - 108. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-471. + 272. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (189. + 328. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-587. - 339. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-346. - 600. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 238. iT - 3.57e5T^{2} \)
73 \( 1 + (-631. + 364. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (439. - 761. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.22e3T + 5.71e5T^{2} \)
89 \( 1 + (-75.9 + 131. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.59e3iT - 9.12e5T^{2} \)
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

−9.677582048467917133718430116012, −8.640410231869594231419008256874, −7.925643480259821312049177678086, −6.92354116910362866512651057631, −6.29363030786821673654615207139, −5.41156790135870868281169245705, −3.96613534211535227091969144285, −3.19250171559101859368772854530, −2.38444811867726460528574847573, −0.47283502906317104117962757696, 0.77364739673583185524858289700, 1.83962536972029506297589538110, 3.69067416088752086994746875435, 4.17277202413388761070546827914, 5.03272846905798502087228116998, 6.35044256523177075039505684177, 7.08294808652716548577148375546, 7.949295477439364901004824413876, 8.937659985366276110530273234754, 9.360416640484535143938356769439

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