↑

Properties

Label 2-3744-39.38-c0-0-9
Degree $2$
Conductor $3744$
Sign $0.816 + 0.577i$
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  + 1.41·5-s − i·13-s − 1.41i·17-s + 1.00·25-s − 1.41i·29-s − 1.41·41-s + 49-s + 1.41i·53-s + 2·61-s − 1.41i·65-s + 2i·73-s − 2.00i·85-s − 1.41·89-s + 2i·97-s − 1.41i·101-s + ⋯
L(s)  = 1  + 1.41·5-s − i·13-s − 1.41i·17-s + 1.00·25-s − 1.41i·29-s − 1.41·41-s + 49-s + 1.41i·53-s + 2·61-s − 1.41i·65-s + 2i·73-s − 2.00i·85-s − 1.41·89-s + 2i·97-s − 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3041, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.654725957\)
\(L(\frac12)\) \(\approx\) \(1.654725957\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - 2iT - T^{2} \)
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

−8.717298675291547585479845286562, −7.891696029393824017686814939047, −7.07265689867623155677646220288, −6.33634260604897680827011216815, −5.51664390314294887398807649123, −5.16302259819073229024949637286, −4.01747593245025883239363820870, −2.82898557991785583771130688416, −2.28565011217361134562623793462, −1.00109666865667709685295600867, 1.56461113898216032464234289170, 2.05292053106648034834927557275, 3.25729397615093100749994637434, 4.18917370526847337885111180948, 5.17925890340391871912981734212, 5.76568350278613785408235613965, 6.59194081367993176915922813907, 6.99886700193495919969474213049, 8.252967581979535579487127644827, 8.791945349429324236578656158242

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