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Properties

Label 2-3744-416.259-c0-0-3
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: $\chi_{3744} (2755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.409845384\)
\(L(\frac12)\) \(\approx\) \(1.409845384\)
\(L(1)\) 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

−8.606310212576718791231063105503, −8.134381663440762684570618569781, −7.02159642482536029484827356972, −5.90323914859699889978292838144, −5.27718482727360884240771889358, −4.87680274116540470390927485098, −3.63057033544026880586946216100, −2.54333233772830022235461195184, −2.01664062260587039899368664837, −0.927971766278985219227596431932, 1.45174902454615745102911940181, 2.44947913099441817966726705212, 3.46519797270171312167136262736, 4.76791786894091134576206406675, 5.39192013345222888509467588548, 6.13661253978331213961533664131, 6.53426949099804350164921158262, 7.27780402683074229330855329063, 8.192524804980343392344077817290, 8.905168177110210653651239837821

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