Properties

Label 2-6045-1.1-c1-0-118
Degree $2$
Conductor $6045$
Sign $1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
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  + 1.15·2-s − 3-s − 0.664·4-s + 5-s − 1.15·6-s + 3.45·7-s − 3.07·8-s + 9-s + 1.15·10-s + 3.68·11-s + 0.664·12-s + 13-s + 3.98·14-s − 15-s − 2.22·16-s + 5.55·17-s + 1.15·18-s + 7.37·19-s − 0.664·20-s − 3.45·21-s + 4.25·22-s − 2.65·23-s + 3.07·24-s + 25-s + 1.15·26-s − 27-s − 2.29·28-s + ⋯
L(s)  = 1  + 0.817·2-s − 0.577·3-s − 0.332·4-s + 0.447·5-s − 0.471·6-s + 1.30·7-s − 1.08·8-s + 0.333·9-s + 0.365·10-s + 1.11·11-s + 0.191·12-s + 0.277·13-s + 1.06·14-s − 0.258·15-s − 0.557·16-s + 1.34·17-s + 0.272·18-s + 1.69·19-s − 0.148·20-s − 0.753·21-s + 0.907·22-s − 0.554·23-s + 0.628·24-s + 0.200·25-s + 0.226·26-s − 0.192·27-s − 0.433·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(48.2695\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.253885130\)
\(L(\frac12)\) \(\approx\) \(3.253885130\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 - T \)
good2 \( 1 - 1.15T + 2T^{2} \)
7 \( 1 - 3.45T + 7T^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
17 \( 1 - 5.55T + 17T^{2} \)
19 \( 1 - 7.37T + 19T^{2} \)
23 \( 1 + 2.65T + 23T^{2} \)
29 \( 1 + 7.40T + 29T^{2} \)
37 \( 1 - 6.10T + 37T^{2} \)
41 \( 1 - 3.82T + 41T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 + 7.44T + 47T^{2} \)
53 \( 1 + 5.58T + 53T^{2} \)
59 \( 1 + 5.32T + 59T^{2} \)
61 \( 1 - 0.0429T + 61T^{2} \)
67 \( 1 - 8.97T + 67T^{2} \)
71 \( 1 - 6.99T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 7.76T + 83T^{2} \)
89 \( 1 - 4.35T + 89T^{2} \)
97 \( 1 - 0.514T + 97T^{2} \)
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

−7.911491130870630838561957629855, −7.40047283104572891274341953580, −6.29596158534258495817448629654, −5.71590313315737598700801148050, −5.27541037524658687807259373477, −4.53261766395410290809320309503, −3.85026548555025878651152233581, −3.05354186245723757566765773438, −1.67044995360846724091084261448, −0.965894180599438061777675938086, 0.965894180599438061777675938086, 1.67044995360846724091084261448, 3.05354186245723757566765773438, 3.85026548555025878651152233581, 4.53261766395410290809320309503, 5.27541037524658687807259373477, 5.71590313315737598700801148050, 6.29596158534258495817448629654, 7.40047283104572891274341953580, 7.911491130870630838561957629855

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