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Properties

Label 2-3920-140.79-c0-0-9
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: $\chi_{3920} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5248849586\)
\(L(\frac12)\) \(\approx\) \(0.5248849586\)
\(L(1)\) 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

−8.378293272833232963978055866766, −7.61581453727304512655077896812, −7.00501857279393429120269258992, −6.04344752980260771131176227772, −5.37285176903902925405164078577, −4.34068765620422180521038197507, −3.80565058961165716497140631245, −2.95879098209780670412631301507, −1.59431164987275454556467023760, −0.28549954460240961735295879634, 1.81962186789723008512524606350, 2.50630904097057243613620254514, 3.78714565461400642748245527060, 4.38458918026463373274329170753, 4.96227370926182661166792839586, 6.22215907784131795899957313471, 7.06740128374258918493847109610, 7.22820608677364547537620961955, 8.166481060483467310752522826861, 9.089526576797051116936218407792

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