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Properties

Label 2-3920-140.39-c0-0-0
Degree $2$
Conductor $3920$
Sign $-0.991 + 0.126i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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 + 1.5i)3-s + (−0.5 − 0.866i)5-s + (−1 − 1.73i)9-s + 1.73·15-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 1.73·27-s − 29-s − 41-s − 1.73·43-s + (−1 + 1.73i)45-s + (0.5 + 0.866i)61-s + (−0.866 + 1.5i)67-s − 3·69-s + (−0.866 − 1.49i)75-s + ⋯
L(s)  = 1  + (−0.866 + 1.5i)3-s + (−0.5 − 0.866i)5-s + (−1 − 1.73i)9-s + 1.73·15-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 1.73·27-s − 29-s − 41-s − 1.73·43-s + (−1 + 1.73i)45-s + (0.5 + 0.866i)61-s + (−0.866 + 1.5i)67-s − 3·69-s + (−0.866 − 1.49i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2919152238\)
\(L(\frac12)\) \(\approx\) \(0.2919152238\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.097987958290444683545724153719, −8.581054020172310062149742119351, −7.59908785334340121379329085271, −6.74396975215330874521149283845, −5.54733266871988576128343171652, −5.37018182768445738903213054815, −4.53442204220787829026252318553, −3.85817765665858858602347762403, −3.18696721227994827156518479338, −1.40622152745667324420238622059, 0.19106114900181863711018993316, 1.58503328259013118929370453294, 2.51428209346487921676847810688, 3.42626318057116166809757937077, 4.64371803828730038681310430530, 5.47604777892725539081538877560, 6.36595439830101845381323828970, 6.76888266107113899432176413754, 7.32273025709275823153683993186, 8.062781919348673036920044042519

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