Properties

Label 2-48e2-24.5-c0-0-3
Degree $2$
Conductor $2304$
Sign $0.985 + 0.169i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·5-s − 1.41i·17-s + 1.00·25-s + 1.41·29-s + 2i·37-s − 1.41i·41-s − 49-s − 1.41·53-s + 2i·61-s − 2.00i·85-s − 1.41i·89-s + 1.41·101-s + 1.41i·113-s + ⋯
L(s)  = 1  + 1.41·5-s − 1.41i·17-s + 1.00·25-s + 1.41·29-s + 2i·37-s − 1.41i·41-s − 49-s − 1.41·53-s + 2i·61-s − 2.00i·85-s − 1.41i·89-s + 1.41·101-s + 1.41i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.985 + 0.169i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.985 + 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.540766485\)
\(L(\frac12)\) \(\approx\) \(1.540766485\)
\(L(1)\) 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 \( 1 - 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + T^{2} \)
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

−9.212266051383144554642996347978, −8.597058472575831558684751570405, −7.56946764151252692614723732508, −6.69382739238078947840478299283, −6.11564829054366919430590158364, −5.19513091173301948970831833379, −4.62177332894830209763602781815, −3.16198802717098525385711424677, −2.41435388271804923174775327567, −1.28015385204437275932220396026, 1.45731166314950487121510487204, 2.29517728673042659054256203652, 3.37968741405111118571776920880, 4.51291407764597439002672536385, 5.37322327612649580110253446526, 6.21368904902907447871323542327, 6.55474490141589343664411811676, 7.80007317043602583412837661978, 8.476116353491173746067148285916, 9.374561146228005112665045471263

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