↑

Properties

Label 2-3744-52.3-c0-0-0
Degree $2$
Conductor $3744$
Sign $0.999 - 0.00641i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (0.866 + 0.5i)7-s + (0.866 − 0.5i)11-s + 13-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)23-s − 25-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + 2i·47-s + (−0.866 − 0.5i)59-s + (0.5 − 0.866i)61-s + (0.866 − 0.5i)67-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.866 − 0.5i)11-s + 13-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.866 + 0.5i)23-s − 25-s + (−0.5 − 0.866i)29-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (−0.866 − 0.5i)43-s + 2i·47-s + (−0.866 − 0.5i)59-s + (0.5 − 0.866i)61-s + (0.866 − 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.999 - 0.00641i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.999 - 0.00641i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.588088359\)
\(L(\frac12)\) \(\approx\) \(1.588088359\)
\(L(1)\) 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 \( 1 \)
13 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.570277533613261015252419535403, −8.004273646191938828477838143945, −7.41040035492513539765480476551, −6.25386161957017360222945412750, −5.78765406991829848564311505319, −5.01946736896274847563390753594, −3.95749065989775475071918108391, −3.36882483790819900265284104356, −2.08527953424548023632994253224, −1.19617245868019273480185179574, 1.25587135189913772951286599086, 1.95606871014051901303345007386, 3.48326567988762892935703111645, 3.97919822360119761812009064347, 4.88046544312526785997962206096, 5.71309792868014311543426564420, 6.50887731512606903965146516735, 7.26376137959833955552672699300, 7.994289831881352483064228469305, 8.600644308696667431970247147020

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