Properties

Label 2-3920-980.779-c0-0-1
Degree $2$
Conductor $3920$
Sign $-0.201 - 0.979i$
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  + (1.64 + 1.12i)3-s + (−0.0747 + 0.997i)5-s + (−0.294 − 0.955i)7-s + (1.08 + 2.77i)9-s + (−1.24 + 1.55i)15-s + (0.587 − 1.90i)21-s + (−0.825 − 0.766i)23-s + (−0.988 − 0.149i)25-s + (−0.877 + 3.84i)27-s + (0.326 + 1.42i)29-s + (0.974 − 0.222i)35-s + (0.658 − 0.317i)41-s + (1.67 + 0.807i)43-s + (−2.84 + 0.877i)45-s + (0.858 − 0.129i)47-s + ⋯
L(s)  = 1  + (1.64 + 1.12i)3-s + (−0.0747 + 0.997i)5-s + (−0.294 − 0.955i)7-s + (1.08 + 2.77i)9-s + (−1.24 + 1.55i)15-s + (0.587 − 1.90i)21-s + (−0.825 − 0.766i)23-s + (−0.988 − 0.149i)25-s + (−0.877 + 3.84i)27-s + (0.326 + 1.42i)29-s + (0.974 − 0.222i)35-s + (0.658 − 0.317i)41-s + (1.67 + 0.807i)43-s + (−2.84 + 0.877i)45-s + (0.858 − 0.129i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.201 - 0.979i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.176477528\)
\(L(\frac12)\) \(\approx\) \(2.176477528\)
\(L(1)\) 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 \( 1 + (0.0747 - 0.997i)T \)
7 \( 1 + (0.294 + 0.955i)T \)
good3 \( 1 + (-1.64 - 1.12i)T + (0.365 + 0.930i)T^{2} \)
11 \( 1 + (0.733 + 0.680i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.825 + 0.766i)T + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (-0.326 - 1.42i)T + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.658 + 0.317i)T + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.67 - 0.807i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.858 + 0.129i)T + (0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (1.57 + 0.487i)T + (0.826 + 0.563i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.367 + 0.460i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.535 + 1.36i)T + (-0.733 + 0.680i)T^{2} \)
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

−9.020192299086310184067222079089, −8.052032601537446755760977982995, −7.58182856977479522152965531851, −6.95561181275955345069021723157, −5.92058624805700952771477817479, −4.61561630781674752539223556386, −4.16644873201317083487665802999, −3.32847525656879219175663049257, −2.86105114794119609747088235342, −1.86620087855593555556577641813, 1.03675281470489166695888994367, 2.08824180465736706134260475848, 2.63835916296264991373810535507, 3.69261921219492304054019069204, 4.37390628717987079958883285074, 5.76553224982843790123057789629, 6.18645856051135221899137224107, 7.32415771843586190538445680453, 7.82489302199118061101274706929, 8.415063451122019332557618031617

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