Properties

Label 2-4840-1.1-c1-0-44
Degree $2$
Conductor $4840$
Sign $1$
Analytic cond. $38.6475$
Root an. cond. $6.21671$
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  − 1.64·3-s + 5-s + 3.37·7-s − 0.306·9-s + 4.71·13-s − 1.64·15-s + 7.55·17-s − 2.22·19-s − 5.54·21-s − 0.0373·23-s + 25-s + 5.42·27-s + 6.36·29-s − 0.348·31-s + 3.37·35-s − 6.63·37-s − 7.73·39-s − 1.97·41-s + 12.1·43-s − 0.306·45-s − 6.36·47-s + 4.40·49-s − 12.4·51-s + 9.06·53-s + 3.65·57-s − 3.10·59-s − 11.6·61-s + ⋯
L(s)  = 1  − 0.947·3-s + 0.447·5-s + 1.27·7-s − 0.102·9-s + 1.30·13-s − 0.423·15-s + 1.83·17-s − 0.511·19-s − 1.20·21-s − 0.00778·23-s + 0.200·25-s + 1.04·27-s + 1.18·29-s − 0.0626·31-s + 0.570·35-s − 1.09·37-s − 1.23·39-s − 0.307·41-s + 1.85·43-s − 0.0457·45-s − 0.929·47-s + 0.628·49-s − 1.73·51-s + 1.24·53-s + 0.484·57-s − 0.404·59-s − 1.49·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(38.6475\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.029979970\)
\(L(\frac12)\) \(\approx\) \(2.029979970\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + 1.64T + 3T^{2} \)
7 \( 1 - 3.37T + 7T^{2} \)
13 \( 1 - 4.71T + 13T^{2} \)
17 \( 1 - 7.55T + 17T^{2} \)
19 \( 1 + 2.22T + 19T^{2} \)
23 \( 1 + 0.0373T + 23T^{2} \)
29 \( 1 - 6.36T + 29T^{2} \)
31 \( 1 + 0.348T + 31T^{2} \)
37 \( 1 + 6.63T + 37T^{2} \)
41 \( 1 + 1.97T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + 6.36T + 47T^{2} \)
53 \( 1 - 9.06T + 53T^{2} \)
59 \( 1 + 3.10T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 - 0.238T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 + 12.0T + 83T^{2} \)
89 \( 1 - 1.87T + 89T^{2} \)
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

−8.339123759702141070986406927944, −7.60684768895947868060048093276, −6.63394259911359115437574838019, −5.94253381533909628495544677925, −5.43371758795748183640079217263, −4.81900127227780956170704271613, −3.86988288233117869963728204617, −2.85434863940879654859994983987, −1.59890858769077248069814917787, −0.913687954964177077015891816505, 0.913687954964177077015891816505, 1.59890858769077248069814917787, 2.85434863940879654859994983987, 3.86988288233117869963728204617, 4.81900127227780956170704271613, 5.43371758795748183640079217263, 5.94253381533909628495544677925, 6.63394259911359115437574838019, 7.60684768895947868060048093276, 8.339123759702141070986406927944

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