↑

Properties

Label 2-3744-39.35-c0-0-2
Degree $2$
Conductor $3744$
Sign $0.935 + 0.352i$
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  − 0.517i·5-s + (0.866 + 0.5i)13-s + (0.448 − 0.258i)17-s + 0.732·25-s + (−0.448 − 0.258i)29-s + (−0.5 + 0.866i)37-s + (0.448 + 0.258i)41-s + (0.5 − 0.866i)49-s − 1.93i·53-s + (0.5 + 0.866i)61-s + (0.258 − 0.448i)65-s + 1.73·73-s + (−0.133 − 0.232i)85-s + (1.22 + 0.707i)89-s + (−1.67 − 0.965i)101-s + ⋯
L(s)  = 1  − 0.517i·5-s + (0.866 + 0.5i)13-s + (0.448 − 0.258i)17-s + 0.732·25-s + (−0.448 − 0.258i)29-s + (−0.5 + 0.866i)37-s + (0.448 + 0.258i)41-s + (0.5 − 0.866i)49-s − 1.93i·53-s + (0.5 + 0.866i)61-s + (0.258 − 0.448i)65-s + 1.73·73-s + (−0.133 − 0.232i)85-s + (1.22 + 0.707i)89-s + (−1.67 − 0.965i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.935 + 0.352i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367721578\)
\(L(\frac12)\) \(\approx\) \(1.367721578\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + 0.517iT - T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.93iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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

−8.566841597925787068443182559766, −8.117036995258154371288812998661, −7.09568424267307842972153972514, −6.49646913598309818144710574255, −5.57756455048760528064670721566, −4.92937790069031592848619222180, −4.02390249490775699308764673380, −3.25919914636652871795207376549, −2.07474879570368458487192460026, −1.00604317038343224894192755749, 1.13040613739690936279843104610, 2.37679593845727478132602435190, 3.30671932163189043116827413005, 3.96783930042591779735696690456, 5.05277178126862492757558209988, 5.83481153028702558758218560667, 6.46347336547139905595207304640, 7.33634512356663185114523906658, 7.908353939765584541469876391018, 8.783618417909722777701795022634

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