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Properties

Label 2-3920-140.39-c0-0-9
Degree $2$
Conductor $3920$
Sign $-0.795 + 0.605i$
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 + (0.866 − 0.5i)5-s + (−1 − 1.73i)9-s + (−1.5 − 0.866i)11-s + i·13-s − 1.73i·15-s + (−0.866 − 0.5i)17-s + (0.499 − 0.866i)25-s − 1.73·27-s + 29-s + (−2.59 + 1.5i)33-s + (1.5 + 0.866i)39-s + (−1.73 − i)45-s + (−0.866 − 1.5i)47-s + (−1.5 + 0.866i)51-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)3-s + (0.866 − 0.5i)5-s + (−1 − 1.73i)9-s + (−1.5 − 0.866i)11-s + i·13-s − 1.73i·15-s + (−0.866 − 0.5i)17-s + (0.499 − 0.866i)25-s − 1.73·27-s + 29-s + (−2.59 + 1.5i)33-s + (1.5 + 0.866i)39-s + (−1.73 − i)45-s + (−0.866 − 1.5i)47-s + (−1.5 + 0.866i)51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.795 + 0.605i$
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: \((2,\ 3920,\ (\ :0),\ -0.795 + 0.605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.694538202\)
\(L(\frac12)\) \(\approx\) \(1.694538202\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
73 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−8.368704153481982314177035336680, −7.75165019069356963831483486029, −6.79459554701298511265282653665, −6.43407794487570624996859376185, −5.48755149614774367319524501951, −4.72186899647891386296479018245, −3.35682207026901733452000509870, −2.43596447516594170361767467658, −2.02642972495275825516799328445, −0.797496923419593349704582391592, 2.08573898517493310998111227735, 2.74401668348127194542872686047, 3.36227566511469527229379191226, 4.51705138939921724877371951478, 5.00045654183884443996016025024, 5.74992068900548257676617220545, 6.71110194433124885020028906512, 7.79831590513079157429554880685, 8.224884821938067648556872374610, 9.122373572224713399006396939166

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