Properties

Label 2-4680-5.4-c1-0-83
Degree $2$
Conductor $4680$
Sign $-0.683 + 0.730i$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
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  + (1.52 − 1.63i)5-s − 2.82i·7-s − 1.05·11-s i·13-s − 7.14i·17-s + 6.61·19-s − 3.49i·23-s + (−0.332 − 4.98i)25-s + 4.82·29-s − 9.65·31-s + (−4.61 − 4.32i)35-s + 8.11i·37-s + 3.26·41-s + 7.44i·43-s − 11.3i·47-s + ⋯
L(s)  = 1  + (0.683 − 0.730i)5-s − 1.06i·7-s − 0.318·11-s − 0.277i·13-s − 1.73i·17-s + 1.51·19-s − 0.728i·23-s + (−0.0664 − 0.997i)25-s + 0.896·29-s − 1.73·31-s + (−0.780 − 0.730i)35-s + 1.33i·37-s + 0.510·41-s + 1.13i·43-s − 1.65i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.683 + 0.730i$
Analytic conductor: \(37.3699\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4680} (2809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ -0.683 + 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.928366458\)
\(L(\frac12)\) \(\approx\) \(1.928366458\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.52 + 1.63i)T \)
13 \( 1 + iT \)
good7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 1.05T + 11T^{2} \)
17 \( 1 + 7.14iT - 17T^{2} \)
19 \( 1 - 6.61T + 19T^{2} \)
23 \( 1 + 3.49iT - 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 - 8.11iT - 37T^{2} \)
41 \( 1 - 3.26T + 41T^{2} \)
43 \( 1 - 7.44iT - 43T^{2} \)
47 \( 1 + 11.3iT - 47T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 + 7.97T + 61T^{2} \)
67 \( 1 - 1.46iT - 67T^{2} \)
71 \( 1 + 0.845T + 71T^{2} \)
73 \( 1 + 9.28iT - 73T^{2} \)
79 \( 1 + 6.85T + 79T^{2} \)
83 \( 1 - 7.97iT - 83T^{2} \)
89 \( 1 - 0.139T + 89T^{2} \)
97 \( 1 + 6.93iT - 97T^{2} \)
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

−7.919594688461571171934702095075, −7.35844061559144918894569242519, −6.69126291915292812367920851214, −5.70663668014923120428059313885, −5.01251807000594883752029422140, −4.52923251221178487018877220867, −3.37633073819091740884116623124, −2.59138011666844579531788559869, −1.31331424913017253034852465512, −0.53720958167864161685784998010, 1.51919696080820127812169142954, 2.26055987208327737256488728166, 3.14623011008978271722492022281, 3.89520460169515740766234321917, 5.17500909928742344101642611359, 5.75699315667655305166563039712, 6.15985132841650751254272630290, 7.21525052107068006473882320706, 7.70689100547020421233568229406, 8.746758902717631829832966882956

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