Properties

Label 2-3744-39.38-c0-0-6
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

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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 \( 1 + 2iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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