Properties

Label 2-4840-1.1-c1-0-81
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  + 0.294·3-s + 5-s − 4.39·7-s − 2.91·9-s + 5.96·13-s + 0.294·15-s − 1.66·17-s + 7.69·19-s − 1.29·21-s − 0.904·23-s + 25-s − 1.74·27-s − 4.73·29-s − 5.12·31-s − 4.39·35-s − 0.184·37-s + 1.75·39-s − 2.62·41-s + 3.59·43-s − 2.91·45-s − 0.776·47-s + 12.2·49-s − 0.491·51-s − 9.59·53-s + 2.26·57-s − 11.2·59-s + 13.1·61-s + ⋯
L(s)  = 1  + 0.170·3-s + 0.447·5-s − 1.65·7-s − 0.970·9-s + 1.65·13-s + 0.0761·15-s − 0.404·17-s + 1.76·19-s − 0.282·21-s − 0.188·23-s + 0.200·25-s − 0.335·27-s − 0.880·29-s − 0.920·31-s − 0.742·35-s − 0.0304·37-s + 0.281·39-s − 0.409·41-s + 0.548·43-s − 0.434·45-s − 0.113·47-s + 1.75·49-s − 0.0688·51-s − 1.31·53-s + 0.300·57-s − 1.46·59-s + 1.68·61-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 \( 1 - 0.294T + 3T^{2} \)
7 \( 1 + 4.39T + 7T^{2} \)
13 \( 1 - 5.96T + 13T^{2} \)
17 \( 1 + 1.66T + 17T^{2} \)
19 \( 1 - 7.69T + 19T^{2} \)
23 \( 1 + 0.904T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + 0.184T + 37T^{2} \)
41 \( 1 + 2.62T + 41T^{2} \)
43 \( 1 - 3.59T + 43T^{2} \)
47 \( 1 + 0.776T + 47T^{2} \)
53 \( 1 + 9.59T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 + 7.79T + 67T^{2} \)
71 \( 1 + 6.97T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 - 4.09T + 83T^{2} \)
89 \( 1 + 0.466T + 89T^{2} \)
97 \( 1 - 7.09T + 97T^{2} \)
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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

−7.942177701179263097673605088904, −7.10695201805045870160613524511, −6.28448879458288857076253430462, −5.89950680602723040559802874398, −5.21539557044887741030676411961, −3.76879395542341523762805375104, −3.37365458068758893890386199553, −2.61821444946614869678694318472, −1.34001673442656258428524432973, 0, 1.34001673442656258428524432973, 2.61821444946614869678694318472, 3.37365458068758893890386199553, 3.76879395542341523762805375104, 5.21539557044887741030676411961, 5.89950680602723040559802874398, 6.28448879458288857076253430462, 7.10695201805045870160613524511, 7.942177701179263097673605088904

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