Properties

Label 2-48e2-8.3-c2-0-62
Degree $2$
Conductor $2304$
Sign $-0.707 + 0.707i$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8i·5-s + 10i·13-s + 16·17-s − 39·25-s − 40i·29-s − 70i·37-s + 80·41-s + 49·49-s − 56i·53-s + 22i·61-s + 80·65-s − 110·73-s − 128i·85-s − 160·89-s − 130·97-s + ⋯
L(s)  = 1  − 1.60i·5-s + 0.769i·13-s + 0.941·17-s − 1.56·25-s − 1.37i·29-s − 1.89i·37-s + 1.95·41-s + 0.999·49-s − 1.05i·53-s + 0.360i·61-s + 1.23·65-s − 1.50·73-s − 1.50i·85-s − 1.79·89-s − 1.34·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.629627728\)
\(L(\frac12)\) \(\approx\) \(1.629627728\)
\(L(2)\) 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 \( 1 + 8iT - 25T^{2} \)
7 \( 1 - 49T^{2} \)
11 \( 1 + 121T^{2} \)
13 \( 1 - 10iT - 169T^{2} \)
17 \( 1 - 16T + 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 40iT - 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 70iT - 1.36e3T^{2} \)
41 \( 1 - 80T + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 56iT - 2.80e3T^{2} \)
59 \( 1 + 3.48e3T^{2} \)
61 \( 1 - 22iT - 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 110T + 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + 160T + 7.92e3T^{2} \)
97 \( 1 + 130T + 9.40e3T^{2} \)
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

−8.580882273943320243697807363457, −7.891538340471918946697573102329, −7.14026767403011227624318095523, −5.89076253622260548545836568733, −5.47037191862805957776267867435, −4.39039331770081936004751706489, −3.96658397178248202279254696339, −2.46254032131779528597604432431, −1.38334480358141592952942846253, −0.42743982385974766447887423793, 1.24912961235045319114499090181, 2.71827680301516099413511440193, 3.11297863139009321289180509152, 4.12110078672504490633348312926, 5.34692039867896409792298054887, 6.05515020503578341482657294307, 6.86598541253982223175523954476, 7.49631161009048434862140078490, 8.153787549790755160341835904355, 9.203115553953619061146510685632

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