Properties

 Label 2-2352-1.1-c3-0-35 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 $0$

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Dirichlet series

 L(s)  = 1 + 3·3-s − 0.726·5-s + 9·9-s + 64.4·11-s − 71.8·13-s − 2.17·15-s − 48.9·17-s + 34.3·19-s + 0.903·23-s − 124.·25-s + 27·27-s + 226.·29-s − 275.·31-s + 193.·33-s + 295.·37-s − 215.·39-s − 186.·41-s + 455.·43-s − 6.53·45-s + 282.·47-s − 146.·51-s + 356.·53-s − 46.8·55-s + 103.·57-s + 729.·59-s − 274.·61-s + 52.1·65-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.0649·5-s + 0.333·9-s + 1.76·11-s − 1.53·13-s − 0.0375·15-s − 0.697·17-s + 0.415·19-s + 0.00818·23-s − 0.995·25-s + 0.192·27-s + 1.45·29-s − 1.59·31-s + 1.02·33-s + 1.31·37-s − 0.884·39-s − 0.710·41-s + 1.61·43-s − 0.0216·45-s + 0.876·47-s − 0.402·51-s + 0.923·53-s − 0.114·55-s + 0.239·57-s + 1.60·59-s − 0.575·61-s + 0.0995·65-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

 $$L(2)$$ $$\approx$$ $$2.838445650$$ $$L(\frac12)$$ $$\approx$$ $$2.838445650$$ $$L(\frac{5}{2})$$ 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 + 0.726T + 125T^{2}$$
11 $$1 - 64.4T + 1.33e3T^{2}$$
13 $$1 + 71.8T + 2.19e3T^{2}$$
17 $$1 + 48.9T + 4.91e3T^{2}$$
19 $$1 - 34.3T + 6.85e3T^{2}$$
23 $$1 - 0.903T + 1.21e4T^{2}$$
29 $$1 - 226.T + 2.43e4T^{2}$$
31 $$1 + 275.T + 2.97e4T^{2}$$
37 $$1 - 295.T + 5.06e4T^{2}$$
41 $$1 + 186.T + 6.89e4T^{2}$$
43 $$1 - 455.T + 7.95e4T^{2}$$
47 $$1 - 282.T + 1.03e5T^{2}$$
53 $$1 - 356.T + 1.48e5T^{2}$$
59 $$1 - 729.T + 2.05e5T^{2}$$
61 $$1 + 274.T + 2.26e5T^{2}$$
67 $$1 + 193.T + 3.00e5T^{2}$$
71 $$1 + 40.5T + 3.57e5T^{2}$$
73 $$1 - 206.T + 3.89e5T^{2}$$
79 $$1 + 937.T + 4.93e5T^{2}$$
83 $$1 - 911.T + 5.71e5T^{2}$$
89 $$1 - 949.T + 7.04e5T^{2}$$
97 $$1 + 39.4T + 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

−8.824640394826639948530446460119, −7.78388801512198031749572621543, −7.15946374303968577534339029030, −6.48072152840715334481716554887, −5.47753945225793074171702242928, −4.38766330391293568108166617855, −3.89600502488203313208177934209, −2.73299720152024619801286306303, −1.90780736966263454895888517634, −0.73130621586554670798405678225, 0.73130621586554670798405678225, 1.90780736966263454895888517634, 2.73299720152024619801286306303, 3.89600502488203313208177934209, 4.38766330391293568108166617855, 5.47753945225793074171702242928, 6.48072152840715334481716554887, 7.15946374303968577534339029030, 7.78388801512198031749572621543, 8.824640394826639948530446460119