Properties

Label 2-4830-1.1-c1-0-19
Degree $2$
Conductor $4830$
Sign $1$
Analytic cond. $38.5677$
Root an. cond. $6.21029$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 1.46·11-s − 12-s + 3.46·13-s − 14-s + 15-s + 16-s − 2·17-s + 18-s + 4·19-s − 20-s + 21-s + 1.46·22-s − 23-s − 24-s + 25-s + 3.46·26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.441·11-s − 0.288·12-s + 0.960·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.312·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.679·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4830\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(38.5677\)
Root analytic conductor: \(6.21029\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4830} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4830,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.334493800\)
\(L(\frac12)\) \(\approx\) \(2.334493800\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 + 0.535T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8.92T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 1.46T + 67T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 4.53T + 89T^{2} \)
97 \( 1 - 7.46T + 97T^{2} \)
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

−8.214854429590917058928578070892, −7.28993252083067400675174726552, −6.69725067817358637713557710299, −6.09658008915516018365370287642, −5.33615983004862243610337478141, −4.58068535907491009304192510202, −3.75187741203651232208766355462, −3.21312982175441756516514056250, −1.92668470134742143102605378909, −0.792773608342001275479717830155, 0.792773608342001275479717830155, 1.92668470134742143102605378909, 3.21312982175441756516514056250, 3.75187741203651232208766355462, 4.58068535907491009304192510202, 5.33615983004862243610337478141, 6.09658008915516018365370287642, 6.69725067817358637713557710299, 7.28993252083067400675174726552, 8.214854429590917058928578070892

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