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Properties

Label 2-3920-140.79-c0-0-4
Degree $2$
Conductor $3920$
Sign $0.126 - 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.126 - 0.991i$
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: \((2,\ 3920,\ (\ :0),\ 0.126 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.641291919\)
\(L(\frac12)\) \(\approx\) \(1.641291919\)
\(L(1)\) 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

−8.863470281164058144540749964247, −8.410059671605904843041228079894, −7.59999495501815697869863599456, −6.64249872765774820952862356646, −5.65676714092821376924725208124, −4.61663693854520706083596093571, −4.27655559800699244905999742179, −3.49810450550787718276406414933, −2.93238948403121787022881030754, −1.33472995810036280332713341613, 0.995031354617777728326785980239, 1.91348192347442417615562820089, 3.00089807536480077262521595967, 3.54086961742850294143380672087, 4.48208332420018558220197383643, 5.82210702507639454374612060835, 6.63092495448071342563383752687, 7.07988781302823446588534541924, 7.71322676547030918159909370536, 8.311630600077863112621928465500

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