Properties

Label 2-3920-1.1-c1-0-23
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 + 3.82·11-s + 3.58·13-s − 0.414·15-s + 6.41·17-s − 3.65·19-s − 0.585·23-s + 25-s − 2.41·27-s + 6.65·29-s − 4.58·31-s + 1.58·33-s − 3.41·37-s + 1.48·39-s + 0.585·41-s − 11.6·43-s + 2.82·45-s + 8.89·47-s + 2.65·51-s − 3.75·53-s − 3.82·55-s − 1.51·57-s − 3.41·59-s + 5.17·61-s − 3.58·65-s + ⋯
L(s)  = 1  + 0.239·3-s − 0.447·5-s − 0.942·9-s + 1.15·11-s + 0.994·13-s − 0.106·15-s + 1.55·17-s − 0.838·19-s − 0.122·23-s + 0.200·25-s − 0.464·27-s + 1.23·29-s − 0.823·31-s + 0.276·33-s − 0.561·37-s + 0.237·39-s + 0.0914·41-s − 1.77·43-s + 0.421·45-s + 1.29·47-s + 0.372·51-s − 0.516·53-s − 0.516·55-s − 0.200·57-s − 0.444·59-s + 0.662·61-s − 0.444·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.963841056\)
\(L(\frac12)\) \(\approx\) \(1.963841056\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.82T + 11T^{2} \)
13 \( 1 - 3.58T + 13T^{2} \)
17 \( 1 - 6.41T + 17T^{2} \)
19 \( 1 + 3.65T + 19T^{2} \)
23 \( 1 + 0.585T + 23T^{2} \)
29 \( 1 - 6.65T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 - 0.585T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 - 8.89T + 47T^{2} \)
53 \( 1 + 3.75T + 53T^{2} \)
59 \( 1 + 3.41T + 59T^{2} \)
61 \( 1 - 5.17T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + 6.48T + 71T^{2} \)
73 \( 1 - 5.17T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.516379132552197252517739679872, −7.919641727185140360795128122359, −6.97023623147530293538794132126, −6.23921338505038665102960687093, −5.61358791471021521469328509118, −4.60753845717234667140618259883, −3.59940902690060265821013856011, −3.27903334769201808688063940427, −1.93637632700613074396246778066, −0.818161780749957320772486021336, 0.818161780749957320772486021336, 1.93637632700613074396246778066, 3.27903334769201808688063940427, 3.59940902690060265821013856011, 4.60753845717234667140618259883, 5.61358791471021521469328509118, 6.23921338505038665102960687093, 6.97023623147530293538794132126, 7.919641727185140360795128122359, 8.516379132552197252517739679872

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