Properties

Label 2-6624-24.11-c1-0-64
Degree $2$
Conductor $6624$
Sign $-0.274 + 0.961i$
Analytic cond. $52.8929$
Root an. cond. $7.27275$
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  − 0.320·5-s + 0.312i·7-s + 4.11i·11-s − 2.83i·13-s − 7.18i·17-s − 0.761·19-s + 23-s − 4.89·25-s + 5.42·29-s + 3.94i·31-s − 0.100i·35-s + 2.91i·37-s − 3.05i·41-s + 7.52·43-s − 8.63·47-s + ⋯
L(s)  = 1  − 0.143·5-s + 0.118i·7-s + 1.24i·11-s − 0.785i·13-s − 1.74i·17-s − 0.174·19-s + 0.208·23-s − 0.979·25-s + 1.00·29-s + 0.709i·31-s − 0.0169i·35-s + 0.478i·37-s − 0.477i·41-s + 1.14·43-s − 1.25·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6624\)    =    \(2^{5} \cdot 3^{2} \cdot 23\)
Sign: $-0.274 + 0.961i$
Analytic conductor: \(52.8929\)
Root analytic conductor: \(7.27275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6624} (4463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6624,\ (\ :1/2),\ -0.274 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.044058423\)
\(L(\frac12)\) \(\approx\) \(1.044058423\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 0.320T + 5T^{2} \)
7 \( 1 - 0.312iT - 7T^{2} \)
11 \( 1 - 4.11iT - 11T^{2} \)
13 \( 1 + 2.83iT - 13T^{2} \)
17 \( 1 + 7.18iT - 17T^{2} \)
19 \( 1 + 0.761T + 19T^{2} \)
29 \( 1 - 5.42T + 29T^{2} \)
31 \( 1 - 3.94iT - 31T^{2} \)
37 \( 1 - 2.91iT - 37T^{2} \)
41 \( 1 + 3.05iT - 41T^{2} \)
43 \( 1 - 7.52T + 43T^{2} \)
47 \( 1 + 8.63T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 5.45iT - 59T^{2} \)
61 \( 1 - 2.02iT - 61T^{2} \)
67 \( 1 - 3.80T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + 3.96T + 73T^{2} \)
79 \( 1 + 6.88iT - 79T^{2} \)
83 \( 1 - 8.44iT - 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + 3.67T + 97T^{2} \)
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

−7.61964742448897313078423453802, −7.23512948987727995910395082092, −6.47106931431341766386872364988, −5.57957029652375594470309748179, −4.85518330528523547849071139386, −4.35111962465586532664882370229, −3.19720404627623296809439807911, −2.58820728043632035570273761402, −1.53651361705918119623990520395, −0.27700796624505484849322890879, 1.08212805423473986488970665902, 2.06000759123255160114010808372, 3.08395370138902679329659191806, 3.92742734007145397145269038994, 4.39726132961038583382340421413, 5.54087391234719450414520481655, 6.14647697781347498777703461449, 6.61121060112843520411034160098, 7.68950092810116833757710684688, 8.153358162109980946803731266876

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