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Properties

Label 2-3744-13.12-c1-0-51
Degree $2$
Conductor $3744$
Sign $0.246 + 0.969i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.49i·5-s − 1.10i·7-s − 3.60i·11-s + (3.49 − 0.890i)13-s − 2·17-s + 1.10i·19-s − 7.20·23-s − 1.21·25-s + 5.20·29-s − 6.09i·31-s + 2.76·35-s − 11.2i·37-s + 7.48i·41-s − 9.42·43-s − 7.60i·47-s + ⋯
L(s)  = 1  + 1.11i·5-s − 0.419i·7-s − 1.08i·11-s + (0.969 − 0.246i)13-s − 0.485·17-s + 0.254i·19-s − 1.50·23-s − 0.243·25-s + 0.967·29-s − 1.09i·31-s + 0.467·35-s − 1.84i·37-s + 1.16i·41-s − 1.43·43-s − 1.10i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.246 + 0.969i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.353426164\)
\(L(\frac12)\) \(\approx\) \(1.353426164\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.49 + 0.890i)T \)
good5 \( 1 - 2.49iT - 5T^{2} \)
7 \( 1 + 1.10iT - 7T^{2} \)
11 \( 1 + 3.60iT - 11T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 1.10iT - 19T^{2} \)
23 \( 1 + 7.20T + 23T^{2} \)
29 \( 1 - 5.20T + 29T^{2} \)
31 \( 1 + 6.09iT - 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 - 7.48iT - 41T^{2} \)
43 \( 1 + 9.42T + 43T^{2} \)
47 \( 1 + 7.60iT - 47T^{2} \)
53 \( 1 + 4.76T + 53T^{2} \)
59 \( 1 - 1.38iT - 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 11.8iT - 67T^{2} \)
71 \( 1 - 1.82iT - 71T^{2} \)
73 \( 1 + 4.98iT - 73T^{2} \)
79 \( 1 + 0.439T + 79T^{2} \)
83 \( 1 + 0.591iT - 83T^{2} \)
89 \( 1 - 1.06iT - 89T^{2} \)
97 \( 1 - 3.01iT - 97T^{2} \)
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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

−8.213613105233331702839128363754, −7.72910253541049051561505468084, −6.71977372014948965989760154332, −6.22850549470489805423251132287, −5.61564891031886363241177288715, −4.32574973363468441518042092488, −3.60922627744535054065707966458, −2.93426832422159201931005458332, −1.84662681430260751888317361621, −0.41146011003596212839999786458, 1.21464108034273529533497536293, 2.01000766163096581666325675280, 3.19750025007155820385308451459, 4.37253291601792498709309867066, 4.70703162858188742041944888054, 5.63972967083310803554410010660, 6.43810421144738405323453685542, 7.13686097603449486018208067773, 8.346537103763153377570107613012, 8.448035707347518548391054584082

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