# Properties

 Label 2-2352-1.1-c3-0-44 Degree $2$ Conductor $2352$ Sign $1$ Analytic cond. $138.772$ Root an. cond. $11.7801$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3·3-s + 18.9·5-s + 9·9-s − 54.7·11-s + 62.0·13-s − 56.8·15-s + 122.·17-s + 12.5·19-s + 74.4·23-s + 234.·25-s − 27·27-s − 232.·29-s − 10.3·31-s + 164.·33-s − 245.·37-s − 186.·39-s + 238.·41-s + 92.9·43-s + 170.·45-s + 485.·47-s − 367.·51-s − 378.·53-s − 1.03e3·55-s − 37.5·57-s + 182.·59-s + 396.·61-s + 1.17e3·65-s + ⋯
 L(s)  = 1 − 0.577·3-s + 1.69·5-s + 0.333·9-s − 1.50·11-s + 1.32·13-s − 0.978·15-s + 1.74·17-s + 0.151·19-s + 0.674·23-s + 1.87·25-s − 0.192·27-s − 1.48·29-s − 0.0600·31-s + 0.866·33-s − 1.09·37-s − 0.763·39-s + 0.909·41-s + 0.329·43-s + 0.565·45-s + 1.50·47-s − 1.00·51-s − 0.981·53-s − 2.54·55-s − 0.0871·57-s + 0.403·59-s + 0.832·61-s + 2.24·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2352$$    =    $$2^{4} \cdot 3 \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$138.772$$ Root analytic conductor: $$11.7801$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{2352} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 2352,\ (\ :3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.878039611$$ $$L(\frac12)$$ $$\approx$$ $$2.878039611$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + 3T$$
7 $$1$$
good5 $$1 - 18.9T + 125T^{2}$$
11 $$1 + 54.7T + 1.33e3T^{2}$$
13 $$1 - 62.0T + 2.19e3T^{2}$$
17 $$1 - 122.T + 4.91e3T^{2}$$
19 $$1 - 12.5T + 6.85e3T^{2}$$
23 $$1 - 74.4T + 1.21e4T^{2}$$
29 $$1 + 232.T + 2.43e4T^{2}$$
31 $$1 + 10.3T + 2.97e4T^{2}$$
37 $$1 + 245.T + 5.06e4T^{2}$$
41 $$1 - 238.T + 6.89e4T^{2}$$
43 $$1 - 92.9T + 7.95e4T^{2}$$
47 $$1 - 485.T + 1.03e5T^{2}$$
53 $$1 + 378.T + 1.48e5T^{2}$$
59 $$1 - 182.T + 2.05e5T^{2}$$
61 $$1 - 396.T + 2.26e5T^{2}$$
67 $$1 - 261.T + 3.00e5T^{2}$$
71 $$1 - 874.T + 3.57e5T^{2}$$
73 $$1 - 152.T + 3.89e5T^{2}$$
79 $$1 + 573.T + 4.93e5T^{2}$$
83 $$1 + 317.T + 5.71e5T^{2}$$
89 $$1 + 95.0T + 7.04e5T^{2}$$
97 $$1 + 1.60e3T + 9.12e5T^{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

−8.739658010500132465782464690833, −7.80504126815167161757594221428, −7.00541759286765564244509733995, −5.95223946637077819540945921463, −5.58642184740769242236728742741, −5.12318318368303619776479128412, −3.67924591047356129973775489867, −2.69293480961935627668399063972, −1.68835392632215501655019969494, −0.817718502839497621070023924337, 0.817718502839497621070023924337, 1.68835392632215501655019969494, 2.69293480961935627668399063972, 3.67924591047356129973775489867, 5.12318318368303619776479128412, 5.58642184740769242236728742741, 5.95223946637077819540945921463, 7.00541759286765564244509733995, 7.80504126815167161757594221428, 8.739658010500132465782464690833