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Properties

Label 2-3744-24.11-c1-0-40
Degree $2$
Conductor $3744$
Sign $0.364 + 0.931i$
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  + 3.97·5-s − 0.281i·7-s − 4.15i·11-s + i·13-s − 5.36i·17-s − 1.44·19-s − 0.283·23-s + 10.7·25-s − 0.381·29-s − 6.98i·31-s − 1.11i·35-s + 2.82i·37-s − 5.37i·41-s − 9.55·43-s − 6.24·47-s + ⋯
L(s)  = 1  + 1.77·5-s − 0.106i·7-s − 1.25i·11-s + 0.277i·13-s − 1.30i·17-s − 0.331·19-s − 0.0590·23-s + 2.15·25-s − 0.0709·29-s − 1.25i·31-s − 0.189i·35-s + 0.464i·37-s − 0.840i·41-s − 1.45·43-s − 0.910·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 0.364 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.518031211\)
\(L(\frac12)\) \(\approx\) \(2.518031211\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 - 3.97T + 5T^{2} \)
7 \( 1 + 0.281iT - 7T^{2} \)
11 \( 1 + 4.15iT - 11T^{2} \)
17 \( 1 + 5.36iT - 17T^{2} \)
19 \( 1 + 1.44T + 19T^{2} \)
23 \( 1 + 0.283T + 23T^{2} \)
29 \( 1 + 0.381T + 29T^{2} \)
31 \( 1 + 6.98iT - 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 + 5.37iT - 41T^{2} \)
43 \( 1 + 9.55T + 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 + 0.416T + 53T^{2} \)
59 \( 1 + 2.68iT - 59T^{2} \)
61 \( 1 + 1.06iT - 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 - 1.24iT - 83T^{2} \)
89 \( 1 + 6.64iT - 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−8.630678702561821239573936304334, −7.60324503098218985294398607258, −6.64703308032456899452297672308, −6.16843150996024057986201157295, −5.43024978949994406863565089480, −4.83317145182477561035328085645, −3.56547187266234746124671722577, −2.65430260254853079927729956094, −1.88977073810392583006234512792, −0.70043345799345788716746401387, 1.47242653415126888400426396249, 2.00494487937846374125377318614, 2.94923208952290497374702936187, 4.14312599924509291883418871319, 5.09085512266736210470397113755, 5.60974843752975681557869750318, 6.51852051861036546505966867929, 6.86727743733944602330983030824, 8.044081918337154912056666374661, 8.727901035428090244528403472508

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