Properties

Label 2-483-161.17-c1-0-27
Degree $2$
Conductor $483$
Sign $0.882 + 0.470i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
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  + (2.45 − 0.234i)2-s + (−0.690 − 0.723i)3-s + (4.00 − 0.772i)4-s + (2.61 + 1.34i)5-s + (−1.86 − 1.61i)6-s + (−0.495 − 2.59i)7-s + (4.92 − 1.44i)8-s + (−0.0475 + 0.998i)9-s + (6.72 + 2.69i)10-s + (−0.297 + 3.12i)11-s + (−3.32 − 2.36i)12-s + (−4.04 + 0.580i)13-s + (−1.82 − 6.26i)14-s + (−0.827 − 2.81i)15-s + (4.16 − 1.66i)16-s + (−2.36 − 6.82i)17-s + ⋯
L(s)  = 1  + (1.73 − 0.165i)2-s + (−0.398 − 0.417i)3-s + (2.00 − 0.386i)4-s + (1.16 + 0.601i)5-s + (−0.760 − 0.659i)6-s + (−0.187 − 0.982i)7-s + (1.74 − 0.511i)8-s + (−0.0158 + 0.332i)9-s + (2.12 + 0.851i)10-s + (−0.0898 + 0.940i)11-s + (−0.959 − 0.683i)12-s + (−1.12 + 0.161i)13-s + (−0.487 − 1.67i)14-s + (−0.213 − 0.727i)15-s + (1.04 − 0.417i)16-s + (−0.572 − 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.882 + 0.470i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.882 + 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.52705 - 0.881769i\)
\(L(\frac12)\) \(\approx\) \(3.52705 - 0.881769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.690 + 0.723i)T \)
7 \( 1 + (0.495 + 2.59i)T \)
23 \( 1 + (-3.52 - 3.24i)T \)
good2 \( 1 + (-2.45 + 0.234i)T + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (-2.61 - 1.34i)T + (2.90 + 4.07i)T^{2} \)
11 \( 1 + (0.297 - 3.12i)T + (-10.8 - 2.08i)T^{2} \)
13 \( 1 + (4.04 - 0.580i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (2.36 + 6.82i)T + (-13.3 + 10.5i)T^{2} \)
19 \( 1 + (2.13 - 6.17i)T + (-14.9 - 11.7i)T^{2} \)
29 \( 1 + (-5.51 + 6.36i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (0.108 + 0.0263i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (-2.39 - 0.114i)T + (36.8 + 3.51i)T^{2} \)
41 \( 1 + (3.67 + 5.72i)T + (-17.0 + 37.2i)T^{2} \)
43 \( 1 + (-0.578 + 1.97i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (7.51 - 4.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.83 - 4.87i)T + (-12.4 - 51.5i)T^{2} \)
59 \( 1 + (1.78 - 4.45i)T + (-42.7 - 40.7i)T^{2} \)
61 \( 1 + (-2.86 - 2.72i)T + (2.90 + 60.9i)T^{2} \)
67 \( 1 + (-4.37 + 3.11i)T + (21.9 - 63.3i)T^{2} \)
71 \( 1 + (4.18 - 9.16i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (0.476 + 2.47i)T + (-67.7 + 27.1i)T^{2} \)
79 \( 1 + (-1.71 - 2.18i)T + (-18.6 + 76.7i)T^{2} \)
83 \( 1 + (-14.7 - 9.48i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (1.77 + 7.30i)T + (-79.1 + 40.7i)T^{2} \)
97 \( 1 + (-4.47 + 2.87i)T + (40.2 - 88.2i)T^{2} \)
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

−11.18168736098979973906343830301, −10.20318494309167455230242867242, −9.683468338338922252118733236955, −7.47119377742940306301897113947, −6.90055774921241639511697271441, −6.15774203864838465509616649577, −5.11763545600270008451209429096, −4.35587748906910598304736158497, −2.85665932680255183114791023749, −1.95190828113005320438686582133, 2.18939721161427679367581665993, 3.21167814205613649807423810245, 4.86498299527906952588777746161, 5.06913360996032838659942436520, 6.22402445352245250960578165812, 6.53984035289949120662233159436, 8.461704118023174979054558683067, 9.275385394653154985270286894396, 10.46386811877355138822176254519, 11.27973585866861055704976970757

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