# 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 $1$

# Related objects

## 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