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Properties

Label 2-3724-76.11-c0-0-2
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: $\chi_{3724} (3431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ -0.211 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.815327552\)
\(L(\frac12)\) \(\approx\) \(2.815327552\)
\(L(1)\) 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

−9.058237094233260455568179815225, −7.86906738510917640599563475639, −7.60795011787031128138602774869, −6.55317951424837402842115975678, −6.02189186281492031888359515432, −4.91036020281776479070697520718, −4.27026675870591517719560576562, −3.68553179156600485277215250780, −2.72956116339424830848870014805, −1.96731395285619897931016149609, 1.16019925302006036032582390610, 2.17293044368975333148887017350, 2.99915762846065234162644832217, 3.62466116095247499080432083322, 4.47272940884091267071342294289, 5.73484887567308191455206168245, 5.91989624513621390697092791723, 6.96323025341566582446910436811, 7.83955196063509924204949394575, 8.496607066947543617901941050601

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