Properties

Label 2-3724-76.7-c0-0-0
Degree $2$
Conductor $3724$
Sign $0.211 - 0.977i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
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  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + 0.999i·8-s + i·11-s − 0.999i·12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 − 0.866i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)24-s + (0.5 − 0.866i)25-s − 0.999i·26-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + 0.999i·8-s + i·11-s − 0.999i·12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 − 0.866i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)24-s + (0.5 − 0.866i)25-s − 0.999i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (2059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.211 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9628632166\)
\(L(\frac12)\) \(\approx\) \(0.9628632166\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
19 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.846223702270906300643758348729, −8.118732980646104323567395382138, −7.38041914464571011958986479968, −6.97773124593274116799975139584, −6.27176493874036219752628276196, −5.05007969614412326335655416324, −4.51989538802498852565503221666, −2.98127608636301815230381888843, −2.23268549516318673663187377952, −1.47060178619330392018016084621, 0.64899369372909015916972719610, 2.24071836350233072084192071369, 2.84615319370463371394031962741, 3.67328250758544461569605996408, 4.32678626327359591705730944546, 5.64799307244947459848592149825, 6.46182308010999440820243151494, 7.36921808307658287797682554423, 8.161697549922354743962751372468, 8.646219146931513665445825257651

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