↑

Properties

Label 2-3024-252.103-c1-0-8
Degree $2$
Conductor $3024$
Sign $0.932 - 0.361i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
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  + (−1.53 + 0.886i)5-s + (−2.64 − 0.0987i)7-s + (−5.33 − 3.08i)11-s + (−4.60 − 2.65i)13-s + (−4.69 + 2.71i)17-s + (0.935 − 1.62i)19-s + (0.562 − 0.324i)23-s + (−0.928 + 1.60i)25-s + (1.14 + 1.98i)29-s + 10.2·31-s + (4.14 − 2.19i)35-s + (1.09 − 1.89i)37-s + (6.64 + 3.83i)41-s + (−1.07 + 0.620i)43-s + 0.468·47-s + ⋯
L(s)  = 1  + (−0.686 + 0.396i)5-s + (−0.999 − 0.0373i)7-s + (−1.60 − 0.929i)11-s + (−1.27 − 0.737i)13-s + (−1.13 + 0.657i)17-s + (0.214 − 0.371i)19-s + (0.117 − 0.0677i)23-s + (−0.185 + 0.321i)25-s + (0.213 + 0.369i)29-s + 1.83·31-s + (0.700 − 0.370i)35-s + (0.180 − 0.311i)37-s + (1.03 + 0.599i)41-s + (−0.163 + 0.0946i)43-s + 0.0683·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.932 - 0.361i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6086429089\)
\(L(\frac12)\) \(\approx\) \(0.6086429089\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2.64 + 0.0987i)T \)
good5 \( 1 + (1.53 - 0.886i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.33 + 3.08i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.60 + 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.69 - 2.71i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.935 + 1.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.562 + 0.324i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.14 - 1.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.64 - 3.83i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.07 - 0.620i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.468T + 47T^{2} \)
53 \( 1 + (-0.941 - 1.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.24T + 59T^{2} \)
61 \( 1 + 6.33iT - 61T^{2} \)
67 \( 1 - 9.93iT - 67T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (4.77 - 2.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + (4.06 + 7.03i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.228 - 0.132i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.5 - 7.26i)T + (48.5 - 84.0i)T^{2} \)
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

−8.603935125061451664772461310294, −7.979509790710904397026015094783, −7.33015232723815699259598071330, −6.57058739769086968950006434149, −5.73175547159679148079240684625, −4.93383699140421930196658559902, −3.97290186568778175550096027899, −2.82808114084507696544226741579, −2.69086286936565058744525277408, −0.50534400096554261240341129702, 0.38351496065811961921914856634, 2.33266912278815929527665996158, 2.77501669060320854075488916213, 4.21508139244432880124233376040, 4.63723014828331485090267658784, 5.50032212716550735640005567744, 6.61597598664476062314711177297, 7.20374557188469089764026763755, 7.85667673976992426227105558123, 8.626623684462106473954743054361

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