# Properties

 Label 2-2352-1.1-c1-0-24 Degree $2$ Conductor $2352$ Sign $-1$ Analytic cond. $18.7808$ Root an. cond. $4.33368$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s − 3.41·5-s + 9-s − 0.828·11-s + 4.24·13-s + 3.41·15-s − 7.41·17-s + 6.82·19-s + 4.82·23-s + 6.65·25-s − 27-s + 2.82·29-s − 2.82·31-s + 0.828·33-s − 1.65·37-s − 4.24·39-s − 10.2·41-s + 11.3·43-s − 3.41·45-s − 4.48·47-s + 7.41·51-s − 2·53-s + 2.82·55-s − 6.82·57-s + 8.48·59-s − 11.0·61-s − 14.4·65-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1.52·5-s + 0.333·9-s − 0.249·11-s + 1.17·13-s + 0.881·15-s − 1.79·17-s + 1.56·19-s + 1.00·23-s + 1.33·25-s − 0.192·27-s + 0.525·29-s − 0.508·31-s + 0.144·33-s − 0.272·37-s − 0.679·39-s − 1.59·41-s + 1.72·43-s − 0.508·45-s − 0.654·47-s + 1.03·51-s − 0.274·53-s + 0.381·55-s − 0.904·57-s + 1.10·59-s − 1.41·61-s − 1.79·65-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: $$18.7808$$ Root analytic conductor: $$4.33368$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{2352} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 2352,\ (\ :1/2),\ -1)$$

## 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$$
3 $$1 + T$$
7 $$1$$
good5 $$1 + 3.41T + 5T^{2}$$
11 $$1 + 0.828T + 11T^{2}$$
13 $$1 - 4.24T + 13T^{2}$$
17 $$1 + 7.41T + 17T^{2}$$
19 $$1 - 6.82T + 19T^{2}$$
23 $$1 - 4.82T + 23T^{2}$$
29 $$1 - 2.82T + 29T^{2}$$
31 $$1 + 2.82T + 31T^{2}$$
37 $$1 + 1.65T + 37T^{2}$$
41 $$1 + 10.2T + 41T^{2}$$
43 $$1 - 11.3T + 43T^{2}$$
47 $$1 + 4.48T + 47T^{2}$$
53 $$1 + 2T + 53T^{2}$$
59 $$1 - 8.48T + 59T^{2}$$
61 $$1 + 11.0T + 61T^{2}$$
67 $$1 + 11.3T + 67T^{2}$$
71 $$1 - 10.4T + 71T^{2}$$
73 $$1 + 7.75T + 73T^{2}$$
79 $$1 + 13.6T + 79T^{2}$$
83 $$1 + 4T + 83T^{2}$$
89 $$1 + 5.75T + 89T^{2}$$
97 $$1 - 0.242T + 97T^{2}$$
show less
$$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

−8.633896624027348567285064966310, −7.72806953602727379737167512769, −7.10276400475580769464366541150, −6.40017318611099806179872290760, −5.32470870015525549358797508304, −4.54388982970115817675053300559, −3.79029730786822254235881330119, −2.93187929445904833014945174614, −1.25671956635564698268715408649, 0, 1.25671956635564698268715408649, 2.93187929445904833014945174614, 3.79029730786822254235881330119, 4.54388982970115817675053300559, 5.32470870015525549358797508304, 6.40017318611099806179872290760, 7.10276400475580769464366541150, 7.72806953602727379737167512769, 8.633896624027348567285064966310