Properties

Label 2-3920-1.1-c1-0-22
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 − 6·11-s − 2·13-s + 15-s + 6·17-s + 8·19-s − 3·23-s + 25-s − 5·27-s + 3·29-s + 2·31-s − 6·33-s + 8·37-s − 2·39-s + 3·41-s − 5·43-s − 2·45-s + 6·51-s + 12·53-s − 6·55-s + 8·57-s + 61-s − 2·65-s + 7·67-s − 3·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.80·11-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 1.83·19-s − 0.625·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.359·31-s − 1.04·33-s + 1.31·37-s − 0.320·39-s + 0.468·41-s − 0.762·43-s − 0.298·45-s + 0.840·51-s + 1.64·53-s − 0.809·55-s + 1.05·57-s + 0.128·61-s − 0.248·65-s + 0.855·67-s − 0.361·69-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

\(L(1)\) \(\approx\) \(2.172796204\)
\(L(\frac12)\) \(\approx\) \(2.172796204\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−8.245594155356374350432576694972, −7.84083580552128825064283430744, −7.31177252768061171345554277405, −6.05619093215691707412482890578, −5.40681579058766526360557602701, −4.97753099540625202482609658322, −3.57202523839656537190128105711, −2.84669687235219822536112075528, −2.30029924108649949531010072972, −0.810374366529333525588588034608, 0.810374366529333525588588034608, 2.30029924108649949531010072972, 2.84669687235219822536112075528, 3.57202523839656537190128105711, 4.97753099540625202482609658322, 5.40681579058766526360557602701, 6.05619093215691707412482890578, 7.31177252768061171345554277405, 7.84083580552128825064283430744, 8.245594155356374350432576694972

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