Properties

Label 2-4840-1.1-c1-0-80
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.484·3-s + 5-s + 0.484·7-s − 2.76·9-s − 5.28·13-s − 0.484·15-s − 2.48·17-s + 4.73·19-s − 0.235·21-s + 4.24·23-s + 25-s + 2.79·27-s + 3.76·29-s − 0.235·31-s + 0.484·35-s + 5.76·37-s + 2.56·39-s + 0.969·41-s + 0.249·43-s − 2.76·45-s + 3.28·47-s − 6.76·49-s + 1.20·51-s + 5.76·53-s − 2.29·57-s − 12.4·59-s − 9.70·61-s + ⋯
L(s)  = 1  − 0.279·3-s + 0.447·5-s + 0.183·7-s − 0.921·9-s − 1.46·13-s − 0.125·15-s − 0.602·17-s + 1.08·19-s − 0.0513·21-s + 0.886·23-s + 0.200·25-s + 0.537·27-s + 0.699·29-s − 0.0422·31-s + 0.0819·35-s + 0.947·37-s + 0.409·39-s + 0.151·41-s + 0.0380·43-s − 0.412·45-s + 0.478·47-s − 0.966·49-s + 0.168·51-s + 0.791·53-s − 0.304·57-s − 1.62·59-s − 1.24·61-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: \(1\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.484T + 3T^{2} \)
7 \( 1 - 0.484T + 7T^{2} \)
13 \( 1 + 5.28T + 13T^{2} \)
17 \( 1 + 2.48T + 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 - 3.76T + 29T^{2} \)
31 \( 1 + 0.235T + 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 - 0.969T + 41T^{2} \)
43 \( 1 - 0.249T + 43T^{2} \)
47 \( 1 - 3.28T + 47T^{2} \)
53 \( 1 - 5.76T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + 5.70T + 71T^{2} \)
73 \( 1 - 1.75T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 - 3.82T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−7.79885204692911857871196854730, −7.23614537708720825789409377286, −6.40365679468772869254277846044, −5.67760441039495144810101383270, −5.00508981389497106931216591502, −4.42204591444375277392075002498, −2.99001427910649527202317956989, −2.61603869146978843024056451333, −1.33206469780861092655692292895, 0, 1.33206469780861092655692292895, 2.61603869146978843024056451333, 2.99001427910649527202317956989, 4.42204591444375277392075002498, 5.00508981389497106931216591502, 5.67760441039495144810101383270, 6.40365679468772869254277846044, 7.23614537708720825789409377286, 7.79885204692911857871196854730

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