↑

Properties

Label 2-2523-87.5-c0-0-3
Degree $2$
Conductor $2523$
Sign $0.944 + 0.328i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (0.360 + 1.57i)7-s + (0.900 − 0.433i)9-s − i·12-s + (0.556 + 0.268i)13-s + (−0.900 − 0.433i)16-s + (−0.602 − 0.137i)19-s + (0.702 + 1.45i)21-s + (−0.222 + 0.974i)25-s + (0.781 − 0.623i)27-s + 1.61·28-s + (1.26 − 1.00i)31-s + (−0.222 − 0.974i)36-s + (0.268 + 0.556i)37-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)3-s + (0.222 − 0.974i)4-s + (0.360 + 1.57i)7-s + (0.900 − 0.433i)9-s − i·12-s + (0.556 + 0.268i)13-s + (−0.900 − 0.433i)16-s + (−0.602 − 0.137i)19-s + (0.702 + 1.45i)21-s + (−0.222 + 0.974i)25-s + (0.781 − 0.623i)27-s + 1.61·28-s + (1.26 − 1.00i)31-s + (−0.222 − 0.974i)36-s + (0.268 + 0.556i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.944 + 0.328i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896154059\)
\(L(\frac12)\) \(\approx\) \(1.896154059\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.974 + 0.222i)T \)
29 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
7 \( 1 + (-0.360 - 1.57i)T + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.556 - 0.268i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.602 + 0.137i)T + (0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (-1.26 + 1.00i)T + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (-0.268 - 0.556i)T + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (1.57 - 0.360i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.556 + 0.268i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (0.483 + 0.385i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 + (0.268 + 0.556i)T + (-0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (1.57 + 0.360i)T + (0.900 + 0.433i)T^{2} \)
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

−8.967809235738652347650844027306, −8.509888437963541001381321158505, −7.66008780337340291038025996551, −6.60559721929723431610654057069, −6.06015284112235875742437785966, −5.19852358257470810404085787702, −4.32350570855367022798887020052, −3.05797101259380730783350651763, −2.21774117122829491258543916756, −1.51110283172942226236013210585, 1.42468404928078472860675137237, 2.66441359374599962459064455548, 3.50850938555838022659232622511, 4.18373609576692480589597194710, 4.74494793862872465053402045432, 6.41969587254055375828543162784, 7.00329417472932272377337253556, 7.901070832844520024600377113920, 8.139610309910418440356872090193, 8.905303543064560405346623516744

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