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Properties

Label 2-3744-416.363-c0-0-0
Degree $2$
Conductor $3744$
Sign $0.382 + 0.923i$
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  + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (−1.81 − 0.750i)5-s + (−0.555 + 0.831i)8-s + (−1.08 + 1.63i)10-s + (1.02 + 0.425i)11-s + (0.382 + 0.923i)13-s + (0.707 + 0.707i)16-s + (1.38 + 1.38i)20-s + (0.617 − 0.923i)22-s + (2.01 + 2.01i)25-s + (0.980 − 0.195i)26-s + (0.831 − 0.555i)32-s + (1.63 − 1.08i)40-s + (0.275 + 0.275i)41-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)2-s + (−0.923 − 0.382i)4-s + (−1.81 − 0.750i)5-s + (−0.555 + 0.831i)8-s + (−1.08 + 1.63i)10-s + (1.02 + 0.425i)11-s + (0.382 + 0.923i)13-s + (0.707 + 0.707i)16-s + (1.38 + 1.38i)20-s + (0.617 − 0.923i)22-s + (2.01 + 2.01i)25-s + (0.980 − 0.195i)26-s + (0.831 − 0.555i)32-s + (1.63 − 1.08i)40-s + (0.275 + 0.275i)41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8797207122\)
\(L(\frac12)\) \(\approx\) \(0.8797207122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.195 + 0.980i)T \)
3 \( 1 \)
13 \( 1 + (-0.382 - 0.923i)T \)
good5 \( 1 + (1.81 + 0.750i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (-1.02 - 0.425i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (-0.275 - 0.275i)T + iT^{2} \)
43 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + 1.66iT - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.149 - 0.360i)T + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.84T + T^{2} \)
83 \( 1 + (-0.750 - 1.81i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
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

−8.633479366323835207300663695766, −8.100465653269144588959010316019, −7.16850636369973617416347397645, −6.36770766953661446435586578171, −5.07308724493962421500936472833, −4.52536934207630248299340884244, −3.84958596710012265607030372670, −3.37878290569971063116149487996, −1.90348298689615791285213196556, −0.879221562805215226194782368553, 0.72447943024161130438743682236, 2.95150869872240870523650685959, 3.64737037359172479579043577275, 4.11544438298024430517535381943, 5.05346532058111392038341934224, 6.09866885350879055693996248065, 6.67525707808097401846060130196, 7.43873436715442819282076846834, 7.896081002430352842071878950685, 8.575597780323810274008584174433

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