Properties

Label 2-3920-1.1-c1-0-6
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·3-s − 5-s − 2.82·9-s + 0.171·11-s − 4.41·13-s + 0.414·15-s − 3.24·17-s + 6·19-s − 7.41·23-s + 25-s + 2.41·27-s − 8.65·29-s + 10.2·31-s − 0.0710·33-s + 2.24·37-s + 1.82·39-s + 6.24·41-s − 2·43-s + 2.82·45-s − 7.24·47-s + 1.34·51-s + 4.24·53-s − 0.171·55-s − 2.48·57-s − 2.24·59-s + 2.82·61-s + 4.41·65-s + ⋯
L(s)  = 1  − 0.239·3-s − 0.447·5-s − 0.942·9-s + 0.0517·11-s − 1.22·13-s + 0.106·15-s − 0.786·17-s + 1.37·19-s − 1.54·23-s + 0.200·25-s + 0.464·27-s − 1.60·29-s + 1.83·31-s − 0.0123·33-s + 0.368·37-s + 0.292·39-s + 0.974·41-s − 0.304·43-s + 0.421·45-s − 1.05·47-s + 0.188·51-s + 0.582·53-s − 0.0231·55-s − 0.329·57-s − 0.291·59-s + 0.362·61-s + 0.547·65-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: no
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9306083289\)
\(L(\frac12)\) \(\approx\) \(0.9306083289\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 3T^{2} \)
11 \( 1 - 0.171T + 11T^{2} \)
13 \( 1 + 4.41T + 13T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 7.41T + 23T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 7.24T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 2.24T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 - 8.24T + 67T^{2} \)
71 \( 1 - 3.17T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + 1.48T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−8.256325891091565577939245090735, −7.85065505292972220561530559851, −7.03679416669897953252115658571, −6.22515159726651672373737081452, −5.45995213551298798063887261836, −4.76043098823771432097158270741, −3.88483953880235867248184423100, −2.92303880235447586509322072964, −2.11145328900084220562022357581, −0.53499513789367496019032163026, 0.53499513789367496019032163026, 2.11145328900084220562022357581, 2.92303880235447586509322072964, 3.88483953880235867248184423100, 4.76043098823771432097158270741, 5.45995213551298798063887261836, 6.22515159726651672373737081452, 7.03679416669897953252115658571, 7.85065505292972220561530559851, 8.256325891091565577939245090735

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