Properties

Label 2-3920-1.1-c1-0-43
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 9-s − 3·11-s + 13-s + 2·15-s + 6·17-s − 19-s − 9·23-s + 25-s + 4·27-s + 6·29-s + 8·31-s + 6·33-s − 7·37-s − 2·39-s − 3·41-s − 2·43-s − 45-s + 9·47-s − 12·51-s + 9·53-s + 3·55-s + 2·57-s − 8·61-s − 65-s − 8·67-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.516·15-s + 1.45·17-s − 0.229·19-s − 1.87·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 1.04·33-s − 1.15·37-s − 0.320·39-s − 0.468·41-s − 0.304·43-s − 0.149·45-s + 1.31·47-s − 1.68·51-s + 1.23·53-s + 0.404·55-s + 0.264·57-s − 1.02·61-s − 0.124·65-s − 0.977·67-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: \(1\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.110665343422627328341703677069, −7.36966289786930473952512070223, −6.44356015470676698239255770103, −5.84902337015059090468915893460, −5.21180825678201850973947638734, −4.45188518883804505234761726142, −3.50107599518529915561545111946, −2.51614662300149095374145418305, −1.09786046285487545053793537670, 0, 1.09786046285487545053793537670, 2.51614662300149095374145418305, 3.50107599518529915561545111946, 4.45188518883804505234761726142, 5.21180825678201850973947638734, 5.84902337015059090468915893460, 6.44356015470676698239255770103, 7.36966289786930473952512070223, 8.110665343422627328341703677069

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