Properties

Label 2-3920-1.1-c1-0-26
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 2·9-s + 3·11-s − 5·13-s + 15-s − 3·17-s + 2·19-s + 6·23-s + 25-s − 5·27-s + 3·29-s − 4·31-s + 3·33-s + 2·37-s − 5·39-s + 12·41-s + 10·43-s − 2·45-s + 9·47-s − 3·51-s + 12·53-s + 3·55-s + 2·57-s − 8·61-s − 5·65-s + 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.904·11-s − 1.38·13-s + 0.258·15-s − 0.727·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s − 0.718·31-s + 0.522·33-s + 0.328·37-s − 0.800·39-s + 1.87·41-s + 1.52·43-s − 0.298·45-s + 1.31·47-s − 0.420·51-s + 1.64·53-s + 0.404·55-s + 0.264·57-s − 1.02·61-s − 0.620·65-s + 0.488·67-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.391079284\)
\(L(\frac12)\) \(\approx\) \(2.391079284\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−8.703677986065848624605724183186, −7.59679103544400835112633947355, −7.18840908028866321022100981317, −6.23576189288857341841080164821, −5.51580216499301883688612135353, −4.66901616031576415432379359350, −3.81894678634232741164227871759, −2.72282955438852544862637314814, −2.28641578958231941287903879491, −0.863841609060871512635709852891, 0.863841609060871512635709852891, 2.28641578958231941287903879491, 2.72282955438852544862637314814, 3.81894678634232741164227871759, 4.66901616031576415432379359350, 5.51580216499301883688612135353, 6.23576189288857341841080164821, 7.18840908028866321022100981317, 7.59679103544400835112633947355, 8.703677986065848624605724183186

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