Properties

Label 2-4840-1.1-c1-0-30
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  − 2.82·3-s − 5-s + 2.09·7-s + 4.99·9-s + 4.99·13-s + 2.82·15-s + 2.80·19-s − 5.92·21-s + 8.45·23-s + 25-s − 5.62·27-s + 4·29-s + 9.18·31-s − 2.09·35-s − 9.98·37-s − 14.1·39-s − 2.53·41-s + 0.233·43-s − 4.99·45-s + 9.15·47-s − 2.61·49-s + 1.33·53-s − 7.91·57-s + 12.1·59-s − 1.12·61-s + 10.4·63-s − 4.99·65-s + ⋯
L(s)  = 1  − 1.63·3-s − 0.447·5-s + 0.791·7-s + 1.66·9-s + 1.38·13-s + 0.729·15-s + 0.642·19-s − 1.29·21-s + 1.76·23-s + 0.200·25-s − 1.08·27-s + 0.742·29-s + 1.64·31-s − 0.354·35-s − 1.64·37-s − 2.25·39-s − 0.395·41-s + 0.0356·43-s − 0.744·45-s + 1.33·47-s − 0.373·49-s + 0.183·53-s − 1.04·57-s + 1.57·59-s − 0.143·61-s + 1.31·63-s − 0.619·65-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

\(L(1)\) \(\approx\) \(1.320000545\)
\(L(\frac12)\) \(\approx\) \(1.320000545\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 - 2.09T + 7T^{2} \)
13 \( 1 - 4.99T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 2.80T + 19T^{2} \)
23 \( 1 - 8.45T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 9.18T + 31T^{2} \)
37 \( 1 + 9.98T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 - 0.233T + 43T^{2} \)
47 \( 1 - 9.15T + 47T^{2} \)
53 \( 1 - 1.33T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 1.12T + 61T^{2} \)
67 \( 1 + 1.50T + 67T^{2} \)
71 \( 1 + 7.72T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 8.60T + 89T^{2} \)
97 \( 1 + 3.04T + 97T^{2} \)
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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

−8.377428793230327453130249405778, −7.27830401907083996696060343487, −6.80284913585645526518369045939, −6.03763807329469101951086607119, −5.28809060233481746525721005045, −4.80020317227107297099797248036, −4.00852298493891587684359885579, −2.98178056527826668803714626857, −1.39194159558798990319283847629, −0.797697770703849508566414565689, 0.797697770703849508566414565689, 1.39194159558798990319283847629, 2.98178056527826668803714626857, 4.00852298493891587684359885579, 4.80020317227107297099797248036, 5.28809060233481746525721005045, 6.03763807329469101951086607119, 6.80284913585645526518369045939, 7.27830401907083996696060343487, 8.377428793230327453130249405778

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