↑

Properties

Label 2-504-21.17-c3-0-6
Degree $2$
Conductor $504$
Sign $0.798 - 0.601i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.798 - 0.601i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ 0.798 - 0.601i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.456313332\)
\(L(\frac12)\) \(\approx\) \(1.456313332\)
\(L(\frac{5}{2})\) 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

−10.84313725616041105247491596066, −9.407446773078699557659264083754, −8.705624139176565691449218146805, −8.180991265680462406133646019625, −7.07977579895064569781095528589, −5.68757212225338359788653929060, −4.91465056060286335402220152929, −4.02313631663255387046648650764, −2.43007066091853244214624097335, −1.02137951138205472004741692778, 0.55977643771031599310019772569, 2.46866277076565371204431455322, 3.49035984226227188972462841385, 4.57213241097385151905058294627, 5.75783660666917880757219228175, 7.11263112415192566189473673129, 7.53389426768535096665305506321, 8.318375610192026067404711823704, 9.859444834194322758012740177597, 10.54135864521159432980944902983

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