↑

Properties

Label 2-3744-104.77-c1-0-49
Degree $2$
Conductor $3744$
Sign $0.738 - 0.673i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
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  + 2.68·5-s + 4.15i·7-s + 4.35·11-s + (3.53 + 0.726i)13-s + 5.87·17-s + 5.71·19-s − 3.62·23-s + 2.23·25-s − 3.08i·29-s + 9.28i·31-s + 11.1i·35-s − 2.69·37-s − 11.1i·41-s − 3.80i·43-s − 4.91i·47-s + ⋯
L(s)  = 1  + 1.20·5-s + 1.56i·7-s + 1.31·11-s + (0.979 + 0.201i)13-s + 1.42·17-s + 1.31·19-s − 0.756·23-s + 0.447·25-s − 0.573i·29-s + 1.66i·31-s + 1.88i·35-s − 0.443·37-s − 1.73i·41-s − 0.580i·43-s − 0.716i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.738 - 0.673i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1585, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 0.738 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.054775539\)
\(L(\frac12)\) \(\approx\) \(3.054775539\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (-3.53 - 0.726i)T \)
good5 \( 1 - 2.68T + 5T^{2} \)
7 \( 1 - 4.15iT - 7T^{2} \)
11 \( 1 - 4.35T + 11T^{2} \)
17 \( 1 - 5.87T + 17T^{2} \)
19 \( 1 - 5.71T + 19T^{2} \)
23 \( 1 + 3.62T + 23T^{2} \)
29 \( 1 + 3.08iT - 29T^{2} \)
31 \( 1 - 9.28iT - 31T^{2} \)
37 \( 1 + 2.69T + 37T^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 + 3.80iT - 43T^{2} \)
47 \( 1 + 4.91iT - 47T^{2} \)
53 \( 1 - 1.17iT - 53T^{2} \)
59 \( 1 - 2.29T + 59T^{2} \)
61 \( 1 + 7.05iT - 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 + 5.13iT - 97T^{2} \)
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

−8.956558040023339098801618485402, −8.012372485506291123964631942109, −6.93901275106249713306017047802, −6.17538299043209839696355654505, −5.63785748175692020019179219480, −5.21988807403456080730268806075, −3.79920687146854160611558887900, −3.07385110144853025043657836487, −1.94654609711017424092161694398, −1.33233089371960447334404965914, 1.14625757179465024749321560761, 1.40674963321736047701679203458, 3.02643040123070856202147331226, 3.78280224998193072033850541555, 4.49074329647984516932636668866, 5.74679913128540017258391970671, 6.02632939180712890931591083387, 6.97619867936133595044190971581, 7.59445171299383064138195524732, 8.381717700613633137622132850982

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