Properties

Label 2-3920-1.1-c1-0-3
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  − 2·3-s + 5-s + 9-s − 3·11-s − 5·13-s − 2·15-s − 6·17-s − 19-s − 3·23-s + 25-s + 4·27-s − 6·29-s − 4·31-s + 6·33-s + 11·37-s + 10·39-s − 3·41-s + 10·43-s + 45-s + 3·47-s + 12·51-s + 3·53-s − 3·55-s + 2·57-s + 4·61-s − 5·65-s + 4·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.38·13-s − 0.516·15-s − 1.45·17-s − 0.229·19-s − 0.625·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 0.718·31-s + 1.04·33-s + 1.80·37-s + 1.60·39-s − 0.468·41-s + 1.52·43-s + 0.149·45-s + 0.437·47-s + 1.68·51-s + 0.412·53-s − 0.404·55-s + 0.264·57-s + 0.512·61-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

\(L(1)\) \(\approx\) \(0.5749970536\)
\(L(\frac12)\) \(\approx\) \(0.5749970536\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.472368949611043666900740569932, −7.52281944204633646242990729706, −6.97700915387173769804896758507, −6.03905383673065348799736760263, −5.59758446718284488917765673392, −4.82572724069122482213651032544, −4.18553147757370413754731582470, −2.69065638848531040226487614455, −2.07540340761280424231662806989, −0.43389262098375283388302574180, 0.43389262098375283388302574180, 2.07540340761280424231662806989, 2.69065638848531040226487614455, 4.18553147757370413754731582470, 4.82572724069122482213651032544, 5.59758446718284488917765673392, 6.03905383673065348799736760263, 6.97700915387173769804896758507, 7.52281944204633646242990729706, 8.472368949611043666900740569932

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