Properties

Label 2-3920-1.1-c1-0-13
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 − 2·13-s − 2·15-s + 6·17-s − 4·19-s − 6·23-s + 25-s + 4·27-s + 6·29-s − 4·31-s + 2·37-s + 4·39-s − 6·41-s + 10·43-s + 45-s − 6·47-s − 12·51-s − 6·53-s + 8·57-s + 12·59-s − 2·61-s − 2·65-s − 2·67-s + 12·69-s + 12·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 1.68·51-s − 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.248·65-s − 0.244·67-s + 1.44·69-s + 1.42·71-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

\(L(1)\) \(\approx\) \(1.067513462\)
\(L(\frac12)\) \(\approx\) \(1.067513462\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.371569178851438535097195720359, −7.72417967231205206440371091917, −6.75938606427748610323373549762, −6.16943207782230729958538690385, −5.54097686482462236146375699951, −4.92936609872116028004359556395, −4.02348966113043943975885714437, −2.91432418996240576588953142404, −1.84751760735400887993660553283, −0.62674328168244276773408798542, 0.62674328168244276773408798542, 1.84751760735400887993660553283, 2.91432418996240576588953142404, 4.02348966113043943975885714437, 4.92936609872116028004359556395, 5.54097686482462236146375699951, 6.16943207782230729958538690385, 6.75938606427748610323373549762, 7.72417967231205206440371091917, 8.371569178851438535097195720359

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