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Properties

Label 2-3744-13.8-c0-0-1
Degree $2$
Conductor $3744$
Sign $0.881 - 0.471i$
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 − i)5-s + i·13-s + 2i·17-s + i·25-s + (1 − i)37-s + (1 + i)41-s + i·49-s + 2·53-s − 2·61-s + (1 − i)65-s + (1 − i)73-s + (2 − 2i)85-s + (1 − i)89-s + (1 + i)97-s + (1 + i)109-s + ⋯
L(s)  = 1  + (−1 − i)5-s + i·13-s + 2i·17-s + i·25-s + (1 − i)37-s + (1 + i)41-s + i·49-s + 2·53-s − 2·61-s + (1 − i)65-s + (1 − i)73-s + (2 − 2i)85-s + (1 − i)89-s + (1 + i)97-s + (1 + i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9583517946\)
\(L(\frac12)\) \(\approx\) \(0.9583517946\)
\(L(1)\) 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 \( 1 + (1 + i)T + iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1 + i)T - iT^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−8.881280032712771804404653911545, −7.905175551103517088204351829065, −7.60879150908012125039689572300, −6.41919343677687892111192996719, −5.86087061692939696009033054717, −4.70647373636620488158144824503, −4.21580925294549505121109840891, −3.57539908031064634315522150312, −2.17349867152280524075897364071, −1.10247418661650765586022074307, 0.66156577215943567323766281094, 2.50902055084983847112073041410, 3.06771659470846232960731048340, 3.89430784817368792590338407325, 4.82838919246430270098992968157, 5.62912378633963076556557223588, 6.60298547932210727145245960736, 7.33639717729686699128713140326, 7.65588509238764072186343001743, 8.518609341742377314456769295243

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