Properties

Label 2-4840-1.1-c1-0-52
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  − 2.74·3-s − 5-s + 0.210·7-s + 4.53·9-s − 2.11·13-s + 2.74·15-s − 2.42·17-s + 4.11·19-s − 0.578·21-s − 5.37·23-s + 25-s − 4.21·27-s + 1.48·29-s + 2.95·31-s − 0.210·35-s + 7.48·37-s + 5.79·39-s − 0.0444·41-s − 10.2·43-s − 4.53·45-s − 7.81·47-s − 6.95·49-s + 6.64·51-s + 12.0·53-s − 11.2·57-s + 9.60·59-s + 8.37·61-s + ⋯
L(s)  = 1  − 1.58·3-s − 0.447·5-s + 0.0796·7-s + 1.51·9-s − 0.585·13-s + 0.708·15-s − 0.587·17-s + 0.943·19-s − 0.126·21-s − 1.12·23-s + 0.200·25-s − 0.810·27-s + 0.276·29-s + 0.530·31-s − 0.0356·35-s + 1.23·37-s + 0.928·39-s − 0.00693·41-s − 1.55·43-s − 0.675·45-s − 1.13·47-s − 0.993·49-s + 0.930·51-s + 1.65·53-s − 1.49·57-s + 1.25·59-s + 1.07·61-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 \( 1 + 2.74T + 3T^{2} \)
7 \( 1 - 0.210T + 7T^{2} \)
13 \( 1 + 2.11T + 13T^{2} \)
17 \( 1 + 2.42T + 17T^{2} \)
19 \( 1 - 4.11T + 19T^{2} \)
23 \( 1 + 5.37T + 23T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 - 2.95T + 31T^{2} \)
37 \( 1 - 7.48T + 37T^{2} \)
41 \( 1 + 0.0444T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 7.81T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 - 9.60T + 59T^{2} \)
61 \( 1 - 8.37T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 9.60T + 83T^{2} \)
89 \( 1 + 5.42T + 89T^{2} \)
97 \( 1 - 5.88T + 97T^{2} \)
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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

−7.83817183505883421768163902530, −6.95092060606248345566448561562, −6.49625732992565874928720883654, −5.67863061798451843207639890762, −5.00249142760864025723263670580, −4.44760871120834793205685070575, −3.51014478496846956764418227637, −2.27389288075158993978811013222, −1.01550164877098077956193324648, 0, 1.01550164877098077956193324648, 2.27389288075158993978811013222, 3.51014478496846956764418227637, 4.44760871120834793205685070575, 5.00249142760864025723263670580, 5.67863061798451843207639890762, 6.49625732992565874928720883654, 6.95092060606248345566448561562, 7.83817183505883421768163902530

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