Properties

Label 2-4840-1.1-c1-0-64
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  − 2.19·3-s + 5-s − 0.544·7-s + 1.81·9-s + 1.63·13-s − 2.19·15-s + 0.169·17-s − 4.72·19-s + 1.19·21-s − 2.26·23-s + 25-s + 2.60·27-s + 9.43·29-s − 3.01·31-s − 0.544·35-s − 0.0537·37-s − 3.59·39-s − 2.54·41-s − 7.46·43-s + 1.81·45-s + 9.07·47-s − 6.70·49-s − 0.370·51-s − 2.03·53-s + 10.3·57-s − 12.6·59-s + 2.80·61-s + ⋯
L(s)  = 1  − 1.26·3-s + 0.447·5-s − 0.205·7-s + 0.603·9-s + 0.454·13-s − 0.566·15-s + 0.0409·17-s − 1.08·19-s + 0.260·21-s − 0.471·23-s + 0.200·25-s + 0.501·27-s + 1.75·29-s − 0.540·31-s − 0.0919·35-s − 0.00882·37-s − 0.575·39-s − 0.397·41-s − 1.13·43-s + 0.270·45-s + 1.32·47-s − 0.957·49-s − 0.0519·51-s − 0.279·53-s + 1.37·57-s − 1.65·59-s + 0.359·61-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 \( 1 + 2.19T + 3T^{2} \)
7 \( 1 + 0.544T + 7T^{2} \)
13 \( 1 - 1.63T + 13T^{2} \)
17 \( 1 - 0.169T + 17T^{2} \)
19 \( 1 + 4.72T + 19T^{2} \)
23 \( 1 + 2.26T + 23T^{2} \)
29 \( 1 - 9.43T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
37 \( 1 + 0.0537T + 37T^{2} \)
41 \( 1 + 2.54T + 41T^{2} \)
43 \( 1 + 7.46T + 43T^{2} \)
47 \( 1 - 9.07T + 47T^{2} \)
53 \( 1 + 2.03T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 2.80T + 61T^{2} \)
67 \( 1 + 1.26T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 7.97T + 73T^{2} \)
79 \( 1 - 3.13T + 79T^{2} \)
83 \( 1 - 2.73T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 7.89T + 97T^{2} \)
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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

−7.967485605488392652288680208259, −6.75142964686878176211938800685, −6.47069503063901706833389475197, −5.81097568073932704657902150934, −5.06154639951441437271598427158, −4.40787320562032348517628675679, −3.37595490781732095499423881163, −2.28612376427139758089142897776, −1.17854084273705461515961225490, 0, 1.17854084273705461515961225490, 2.28612376427139758089142897776, 3.37595490781732095499423881163, 4.40787320562032348517628675679, 5.06154639951441437271598427158, 5.81097568073932704657902150934, 6.47069503063901706833389475197, 6.75142964686878176211938800685, 7.967485605488392652288680208259

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