↑

Properties

Label 2-3744-24.11-c1-0-44
Degree $2$
Conductor $3744$
Sign $-0.122 + 0.992i$
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  + 3.68·5-s − 5.03i·7-s − 3.64i·11-s − i·13-s − 5.67i·17-s + 1.84·19-s + 2.33·23-s + 8.55·25-s + 4.56·29-s + 10.2i·31-s − 18.5i·35-s + 1.61i·37-s + 11.3i·41-s + 1.54·43-s − 7.17·47-s + ⋯
L(s)  = 1  + 1.64·5-s − 1.90i·7-s − 1.09i·11-s − 0.277i·13-s − 1.37i·17-s + 0.422·19-s + 0.487·23-s + 1.71·25-s + 0.846·29-s + 1.83i·31-s − 3.13i·35-s + 0.265i·37-s + 1.76i·41-s + 0.235·43-s − 1.04·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ -0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.650264520\)
\(L(\frac12)\) \(\approx\) \(2.650264520\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good5 \( 1 - 3.68T + 5T^{2} \)
7 \( 1 + 5.03iT - 7T^{2} \)
11 \( 1 + 3.64iT - 11T^{2} \)
17 \( 1 + 5.67iT - 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 - 1.54T + 43T^{2} \)
47 \( 1 + 7.17T + 47T^{2} \)
53 \( 1 + 2.41T + 53T^{2} \)
59 \( 1 + 7.69iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + 6.50T + 71T^{2} \)
73 \( 1 - 7.47T + 73T^{2} \)
79 \( 1 + 5.87iT - 79T^{2} \)
83 \( 1 - 1.25iT - 83T^{2} \)
89 \( 1 - 1.11iT - 89T^{2} \)
97 \( 1 - 6.82T + 97T^{2} \)
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

−8.301385410272078346605064794122, −7.50515916749371792622151010663, −6.62150121392191831282590081499, −6.35510213240364314710027503769, −5.08258682320820602431913812631, −4.84460146930245506850993522113, −3.39585845800066520809966417804, −2.90410078511341018209994742508, −1.44474970626762148145870828473, −0.78059398758596737545152188957, 1.64522967688439301027456637940, 2.16748370368293125619679026203, 2.84202911447716918151480097278, 4.26466325443297270825350276006, 5.22874686627546661430123198284, 5.77774590960386503366871601812, 6.21942720197616959931099111927, 7.07814958365184895851436605822, 8.184440525614596845081121914259, 8.907017433223192809353024083971

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