Properties

Label 2-3920-1.1-c1-0-21
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  + 5-s − 3·9-s − 4·11-s + 6·13-s − 2·17-s + 25-s + 6·29-s + 8·31-s − 10·37-s − 2·41-s − 4·43-s − 3·45-s + 8·47-s − 2·53-s − 4·55-s − 8·59-s + 14·61-s + 6·65-s + 12·67-s + 16·71-s − 2·73-s + 8·79-s + 9·81-s + 8·83-s − 2·85-s − 10·89-s − 2·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 9-s − 1.20·11-s + 1.66·13-s − 0.485·17-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.64·37-s − 0.312·41-s − 0.609·43-s − 0.447·45-s + 1.16·47-s − 0.274·53-s − 0.539·55-s − 1.04·59-s + 1.79·61-s + 0.744·65-s + 1.46·67-s + 1.89·71-s − 0.234·73-s + 0.900·79-s + 81-s + 0.878·83-s − 0.216·85-s − 1.05·89-s − 0.203·97-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

\(L(1)\) \(\approx\) \(1.784452441\)
\(L(\frac12)\) \(\approx\) \(1.784452441\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−8.400283535029584108147575940934, −8.058922554914412122379413087558, −6.78248070030119110364171611800, −6.28136292817627051453406742773, −5.47254939185735647334240808646, −4.90054686617114529340998403638, −3.72179923773102648718320811330, −2.92257849808604956447035538813, −2.09663760025842118849257213691, −0.75960958349138602912815032456, 0.75960958349138602912815032456, 2.09663760025842118849257213691, 2.92257849808604956447035538813, 3.72179923773102648718320811330, 4.90054686617114529340998403638, 5.47254939185735647334240808646, 6.28136292817627051453406742773, 6.78248070030119110364171611800, 8.058922554914412122379413087558, 8.400283535029584108147575940934

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