Properties

Label 2-4840-1.1-c1-0-11
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  − 2.90·3-s + 5-s + 2.18·7-s + 5.43·9-s − 6.16·13-s − 2.90·15-s − 7.12·17-s − 6.35·19-s − 6.34·21-s + 2.17·23-s + 25-s − 7.08·27-s + 3.19·29-s + 4.39·31-s + 2.18·35-s + 5.29·37-s + 17.8·39-s − 3.28·41-s + 6.83·43-s + 5.43·45-s − 4.30·47-s − 2.23·49-s + 20.7·51-s − 10.6·53-s + 18.4·57-s + 13.7·59-s − 7.35·61-s + ⋯
L(s)  = 1  − 1.67·3-s + 0.447·5-s + 0.825·7-s + 1.81·9-s − 1.70·13-s − 0.750·15-s − 1.72·17-s − 1.45·19-s − 1.38·21-s + 0.454·23-s + 0.200·25-s − 1.36·27-s + 0.592·29-s + 0.789·31-s + 0.369·35-s + 0.870·37-s + 2.86·39-s − 0.513·41-s + 1.04·43-s + 0.810·45-s − 0.627·47-s − 0.319·49-s + 2.89·51-s − 1.45·53-s + 2.44·57-s + 1.79·59-s − 0.941·61-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

\(L(1)\) \(\approx\) \(0.7630940270\)
\(L(\frac12)\) \(\approx\) \(0.7630940270\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.90T + 3T^{2} \)
7 \( 1 - 2.18T + 7T^{2} \)
13 \( 1 + 6.16T + 13T^{2} \)
17 \( 1 + 7.12T + 17T^{2} \)
19 \( 1 + 6.35T + 19T^{2} \)
23 \( 1 - 2.17T + 23T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 - 4.39T + 31T^{2} \)
37 \( 1 - 5.29T + 37T^{2} \)
41 \( 1 + 3.28T + 41T^{2} \)
43 \( 1 - 6.83T + 43T^{2} \)
47 \( 1 + 4.30T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 + 7.35T + 61T^{2} \)
67 \( 1 + 7.67T + 67T^{2} \)
71 \( 1 - 5.74T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 - 8.11T + 79T^{2} \)
83 \( 1 + 4.22T + 83T^{2} \)
89 \( 1 - 16.8T + 89T^{2} \)
97 \( 1 + 3.27T + 97T^{2} \)
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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

−8.190203278888162715474667599286, −7.33880430227285786555781492088, −6.51692006496618885944381963493, −6.29421050865654906853954336968, −5.16885457024742393920711577678, −4.73187030169045684492707735112, −4.33451746826290823119039642656, −2.55900537997964834910499636542, −1.82875798800132782955081270674, −0.51205311569964991798862649159, 0.51205311569964991798862649159, 1.82875798800132782955081270674, 2.55900537997964834910499636542, 4.33451746826290823119039642656, 4.73187030169045684492707735112, 5.16885457024742393920711577678, 6.29421050865654906853954336968, 6.51692006496618885944381963493, 7.33880430227285786555781492088, 8.190203278888162715474667599286

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