Properties

Label 2-4840-1.1-c1-0-106
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  + 1.19·3-s + 5-s + 1.83·7-s − 1.57·9-s + 1.25·13-s + 1.19·15-s − 6.44·17-s − 1.25·19-s + 2.19·21-s − 4.35·23-s + 25-s − 5.46·27-s + 0.726·29-s − 11.0·31-s + 1.83·35-s − 11.5·37-s + 1.49·39-s − 7.92·41-s − 0.606·43-s − 1.57·45-s − 3.98·47-s − 3.62·49-s − 7.69·51-s + 6.83·53-s − 1.49·57-s − 1.41·59-s + 9.75·61-s + ⋯
L(s)  = 1  + 0.689·3-s + 0.447·5-s + 0.694·7-s − 0.525·9-s + 0.348·13-s + 0.308·15-s − 1.56·17-s − 0.287·19-s + 0.478·21-s − 0.908·23-s + 0.200·25-s − 1.05·27-s + 0.134·29-s − 1.98·31-s + 0.310·35-s − 1.89·37-s + 0.239·39-s − 1.23·41-s − 0.0925·43-s − 0.234·45-s − 0.580·47-s − 0.517·49-s − 1.07·51-s + 0.938·53-s − 0.197·57-s − 0.183·59-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 \( 1 - 1.19T + 3T^{2} \)
7 \( 1 - 1.83T + 7T^{2} \)
13 \( 1 - 1.25T + 13T^{2} \)
17 \( 1 + 6.44T + 17T^{2} \)
19 \( 1 + 1.25T + 19T^{2} \)
23 \( 1 + 4.35T + 23T^{2} \)
29 \( 1 - 0.726T + 29T^{2} \)
31 \( 1 + 11.0T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + 7.92T + 41T^{2} \)
43 \( 1 + 0.606T + 43T^{2} \)
47 \( 1 + 3.98T + 47T^{2} \)
53 \( 1 - 6.83T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 9.75T + 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 6.79T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 6.51T + 79T^{2} \)
83 \( 1 + 0.644T + 83T^{2} \)
89 \( 1 - 9.39T + 89T^{2} \)
97 \( 1 + 4.66T + 97T^{2} \)
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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

−8.243462533350451460082492058444, −7.16718329099270276182793892501, −6.57582476797937640357036760897, −5.62314212656822501058979812387, −5.04810768668224938376351018351, −4.02506768314878874099110271242, −3.34684692209378124286060059529, −2.15311054834223676303569530227, −1.82435628428542619986078409109, 0, 1.82435628428542619986078409109, 2.15311054834223676303569530227, 3.34684692209378124286060059529, 4.02506768314878874099110271242, 5.04810768668224938376351018351, 5.62314212656822501058979812387, 6.57582476797937640357036760897, 7.16718329099270276182793892501, 8.243462533350451460082492058444

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