Properties

Label 2-3920-35.19-c0-0-1
Degree $2$
Conductor $3920$
Sign $-0.895 - 0.444i$
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)11-s − 13-s + 0.999·15-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)25-s − 27-s − 29-s + (−0.499 + 0.866i)33-s + (0.5 + 0.866i)39-s + (−0.5 + 0.866i)47-s + (0.499 − 0.866i)51-s + 0.999·55-s + (0.5 − 0.866i)65-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)11-s − 13-s + 0.999·15-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)25-s − 27-s − 29-s + (−0.499 + 0.866i)33-s + (0.5 + 0.866i)39-s + (−0.5 + 0.866i)47-s + (0.499 − 0.866i)51-s + 0.999·55-s + (0.5 − 0.866i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04042384498\)
\(L(\frac12)\) \(\approx\) \(0.04042384498\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−7.86519428439525497819822141311, −7.53005861239001141794397994772, −6.80739465675325071096302125763, −6.06115396146323053010651760243, −5.54640364137133478784855125364, −4.35521299984307247806561806836, −3.44833510552664789025519452950, −2.65682932067581481402282493049, −1.51996097849080123173746990361, −0.02366696605190162706817059499, 1.69810432532893621843676467960, 2.87194927327832905627592696237, 4.04345882523985575165827446863, 4.60341898679440797292524763379, 5.21017622692175336726612236322, 5.70177951799821100173811433407, 7.24501194664841360452335092537, 7.40267394829386834699338797138, 8.372765247895775922316789846984, 9.208004666237471646548035870310

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