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Properties

Label 2-3744-39.29-c0-0-3
Degree $2$
Conductor $3744$
Sign $-0.935 + 0.352i$
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  − 1.93i·5-s + (−0.866 + 0.5i)13-s + (−1.67 − 0.965i)17-s − 2.73·25-s + (1.67 − 0.965i)29-s + (−0.5 − 0.866i)37-s + (−1.67 + 0.965i)41-s + (0.5 + 0.866i)49-s − 0.517i·53-s + (0.5 − 0.866i)61-s + (0.965 + 1.67i)65-s − 1.73·73-s + (−1.86 + 3.23i)85-s + (1.22 − 0.707i)89-s + (0.448 − 0.258i)101-s + ⋯
L(s)  = 1  − 1.93i·5-s + (−0.866 + 0.5i)13-s + (−1.67 − 0.965i)17-s − 2.73·25-s + (1.67 − 0.965i)29-s + (−0.5 − 0.866i)37-s + (−1.67 + 0.965i)41-s + (0.5 + 0.866i)49-s − 0.517i·53-s + (0.5 − 0.866i)61-s + (0.965 + 1.67i)65-s − 1.73·73-s + (−1.86 + 3.23i)85-s + (1.22 − 0.707i)89-s + (0.448 − 0.258i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.935 + 0.352i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3617, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7462490222\)
\(L(\frac12)\) \(\approx\) \(0.7462490222\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + 1.93iT - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.517iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.561425385036472738949770024932, −7.77728291223627006059419659356, −6.90162616557042825076348941528, −6.09732497377764918826704231171, −4.96877949467002720865911066917, −4.78013181099185358843383794621, −4.04160398887969563654994681259, −2.59401780935714606018426831648, −1.68842767772875043835417660200, −0.39601022560456549636334922861, 1.95290697473395331420807836470, 2.71814478219887704315846168782, 3.41716736308766273314432241152, 4.35175655603242967950099722623, 5.34529943208513139756876375938, 6.36311200214854983609024620318, 6.78328229648982610822605271319, 7.31197303796858105014103456428, 8.262122315155284735843091722615, 8.905574880052549918960648633554

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