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Properties

Label 2-3744-416.155-c0-0-0
Degree $2$
Conductor $3744$
Sign $-0.923 - 0.382i$
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  + (−0.555 − 0.831i)2-s + (−0.382 + 0.923i)4-s + (−0.636 + 1.53i)5-s + (0.980 − 0.195i)8-s + (1.63 − 0.324i)10-s + (−0.750 + 1.81i)11-s + (−0.923 + 0.382i)13-s + (−0.707 − 0.707i)16-s + (−1.17 − 1.17i)20-s + (1.92 − 0.382i)22-s + (−1.24 − 1.24i)25-s + (0.831 + 0.555i)26-s + (−0.195 + 0.980i)32-s + (−0.324 + 1.63i)40-s + (0.785 + 0.785i)41-s + ⋯
L(s)  = 1  + (−0.555 − 0.831i)2-s + (−0.382 + 0.923i)4-s + (−0.636 + 1.53i)5-s + (0.980 − 0.195i)8-s + (1.63 − 0.324i)10-s + (−0.750 + 1.81i)11-s + (−0.923 + 0.382i)13-s + (−0.707 − 0.707i)16-s + (−1.17 − 1.17i)20-s + (1.92 − 0.382i)22-s + (−1.24 − 1.24i)25-s + (0.831 + 0.555i)26-s + (−0.195 + 0.980i)32-s + (−0.324 + 1.63i)40-s + (0.785 + 0.785i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1819, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3430248027\)
\(L(\frac12)\) \(\approx\) \(0.3430248027\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.555 + 0.831i)T \)
3 \( 1 \)
13 \( 1 + (0.923 - 0.382i)T \)
good5 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.750 - 1.81i)T + (-0.707 - 0.707i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.785 - 0.785i)T + iT^{2} \)
43 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 - 0.390iT - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (1.02 + 0.425i)T + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 0.765T + T^{2} \)
83 \( 1 + (1.53 - 0.636i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−9.348993834067988961741542144388, −8.094632957259958252077914566031, −7.47683143728997523625555721777, −7.22289009223178612685546811617, −6.35763176766939321363426086721, −4.86505048178185633143481815571, −4.33789959827821467532097427619, −3.31561749540502469378603694402, −2.56903999892363756354564307375, −1.93858071309876101316100778621, 0.25654025324044927180753953098, 1.19078768396284301261183835688, 2.75190196958738392224628625667, 4.03400900698331486533177889908, 4.76309207113673272728842651505, 5.58794181659207251968235745116, 5.85072018063720104696857558836, 7.18824343427221549512172110582, 7.81706885094191042132668368089, 8.329786165043332903335963720038

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