Properties

Label 2-4608-24.11-c1-0-49
Degree $2$
Conductor $4608$
Sign $0.577 + 0.816i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  + 3.69·5-s − 3.41i·7-s − 2.16i·11-s − 2.16i·13-s + 1.41i·17-s + 7.39·19-s + 4.82·23-s + 8.65·25-s − 6.75·29-s + 2.24i·31-s − 12.6i·35-s + 7.39i·37-s + 12.2i·41-s + 10.4·43-s − 3.17·47-s + ⋯
L(s)  = 1  + 1.65·5-s − 1.29i·7-s − 0.652i·11-s − 0.600i·13-s + 0.342i·17-s + 1.69·19-s + 1.00·23-s + 1.73·25-s − 1.25·29-s + 0.402i·31-s − 2.13i·35-s + 1.21i·37-s + 1.91i·41-s + 1.59·43-s − 0.462·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.976616938\)
\(L(\frac12)\) \(\approx\) \(2.976616938\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 + 3.41iT - 7T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 + 2.16iT - 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 - 7.39T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 6.75T + 29T^{2} \)
31 \( 1 - 2.24iT - 31T^{2} \)
37 \( 1 - 7.39iT - 37T^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 3.17T + 47T^{2} \)
53 \( 1 - 6.75T + 53T^{2} \)
59 \( 1 + 10.4iT - 59T^{2} \)
61 \( 1 + 3.06iT - 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 - 6.48T + 71T^{2} \)
73 \( 1 - 3.65T + 73T^{2} \)
79 \( 1 + 17.0iT - 79T^{2} \)
83 \( 1 - 8.28iT - 83T^{2} \)
89 \( 1 + 7.07iT - 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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

−8.145708985645595883114418101554, −7.42150671772310507678500096703, −6.72960402869696341035305753823, −5.95500271343394620203013285078, −5.37694519095316454406494541028, −4.63856481422225212670564583307, −3.42642078053303755143018085869, −2.87697194798294363656192014435, −1.54428046636117024254612366167, −0.908110282712652384243857631432, 1.25317224035400644878349259476, 2.26221791983709642465466451753, 2.60046963557257122331755587858, 3.88112313368845467249860424301, 5.13426364389626945820546834854, 5.50055186617722065451823616165, 5.99092587327837417667523370000, 7.03999216868491185876425931921, 7.47399211406665416370861623269, 8.840414601286037516285881866185

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