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Properties

Label 2-3920-140.39-c0-0-6
Degree $2$
Conductor $3920$
Sign $0.922 + 0.386i$
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 − 0.5i)5-s + (0.5 + 0.866i)9-s − 2i·13-s + (1.73 + i)17-s + (0.499 − 0.866i)25-s − 2·29-s + (0.866 + 0.499i)45-s + (−1 − 1.73i)65-s + (1.73 + i)73-s + (−0.499 + 0.866i)81-s + 1.99·85-s − 2i·97-s + (−1 + 1.73i)109-s + (1.73 − i)117-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + (0.5 + 0.866i)9-s − 2i·13-s + (1.73 + i)17-s + (0.499 − 0.866i)25-s − 2·29-s + (0.866 + 0.499i)45-s + (−1 − 1.73i)65-s + (1.73 + i)73-s + (−0.499 + 0.866i)81-s + 1.99·85-s − 2i·97-s + (−1 + 1.73i)109-s + (1.73 − i)117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.922 + 0.386i$
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: \((2,\ 3920,\ (\ :0),\ 0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.655004245\)
\(L(\frac12)\) \(\approx\) \(1.655004245\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 2T + T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 2iT - T^{2} \)
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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

−8.418574673834591410736381274313, −7.917996879118069482524640694649, −7.36039972503599067515737213706, −6.13807167678287230776968203034, −5.44023125587389568714824232883, −5.23526744417419994762963215355, −3.96959647685308557387790409369, −3.10337384369007198507850649765, −2.03367807888651183556892160943, −1.11831237650297407011394469092, 1.33107996145596575114132561806, 2.16833957810339294480032665375, 3.30912613006016697680267459622, 3.98058925832195806117775795481, 5.04314864506762778018709152836, 5.79345676558268074906752727867, 6.58330617910361011557887571935, 7.08807659222138977817010751552, 7.76691092711698886883391325341, 9.071298528772238584678727792450

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