↑

Properties

Label 2-3744-39.23-c0-0-0
Degree $2$
Conductor $3744$
Sign $0.406 + 0.913i$
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  − 1.93·5-s + (−0.866 + 0.5i)13-s + (0.448 + 0.258i)17-s + 2.73·25-s + (−0.448 + 0.258i)29-s + (1.5 − 0.866i)37-s + (−0.965 − 1.67i)41-s + (−0.5 − 0.866i)49-s − 1.93i·53-s + (0.5 − 0.866i)61-s + (1.67 − 0.965i)65-s − i·73-s + (−0.866 − 0.499i)85-s + (0.707 + 1.22i)89-s + (−1.73 − i)97-s + ⋯
L(s)  = 1  − 1.93·5-s + (−0.866 + 0.5i)13-s + (0.448 + 0.258i)17-s + 2.73·25-s + (−0.448 + 0.258i)29-s + (1.5 − 0.866i)37-s + (−0.965 − 1.67i)41-s + (−0.5 − 0.866i)49-s − 1.93i·53-s + (0.5 − 0.866i)61-s + (1.67 − 0.965i)65-s − i·73-s + (−0.866 − 0.499i)85-s + (0.707 + 1.22i)89-s + (−1.73 − i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.406 + 0.913i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.406 + 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6134785262\)
\(L(\frac12)\) \(\approx\) \(0.6134785262\)
\(L(1)\) 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 \( 1 + 1.93T + T^{2} \)
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 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
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

−8.398437758996965241519379621670, −7.79549252889581947039103236279, −7.22170045328173722531520987867, −6.61358300065044925919553261770, −5.36359715763413254867407919794, −4.65856936448082358543215606803, −3.85412034089259137575129601498, −3.32289677600560168776593942113, −2.09008907314902360738791821798, −0.44435091044703936769839552905, 0.999996956377009450436392673061, 2.75638218972090725794355030397, 3.32482396118400615315016909200, 4.36872456174880902339675259346, 4.74284429635340657471318555727, 5.84147077869605373032378110097, 6.85338973484458485218776180263, 7.56497826406676865796067416347, 7.923077507039192574155985896507, 8.593441970219817798481818993730

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