# Properties

 Label 2-96e2-1.1-c1-0-105 Degree $2$ Conductor $9216$ Sign $-1$ Analytic cond. $73.5901$ Root an. cond. $8.57846$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 0.473·5-s − 4.55·7-s + 3.49·11-s − 0.0840·13-s + 3.61·17-s − 3.61·19-s + 2.82·23-s − 4.77·25-s − 7.30·29-s + 0.557·31-s − 2.15·35-s + 6.20·37-s + 9.27·41-s − 2.27·43-s − 2.82·47-s + 13.7·49-s − 0.697·53-s + 1.65·55-s + 5.65·59-s − 3.85·61-s − 0.0397·65-s + 5.33·67-s − 9.11·71-s − 0.541·73-s − 15.9·77-s − 10.9·79-s − 15.0·83-s + ⋯
 L(s)  = 1 + 0.211·5-s − 1.72·7-s + 1.05·11-s − 0.0233·13-s + 0.877·17-s − 0.829·19-s + 0.589·23-s − 0.955·25-s − 1.35·29-s + 0.100·31-s − 0.364·35-s + 1.01·37-s + 1.44·41-s − 0.347·43-s − 0.412·47-s + 1.96·49-s − 0.0958·53-s + 0.223·55-s + 0.736·59-s − 0.494·61-s − 0.00493·65-s + 0.652·67-s − 1.08·71-s − 0.0633·73-s − 1.81·77-s − 1.23·79-s − 1.65·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9216$$    =    $$2^{10} \cdot 3^{2}$$ Sign: $-1$ Analytic conductor: $$73.5901$$ Root analytic conductor: $$8.57846$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{9216} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9216,\ (\ :1/2),\ -1)$$

## 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$$
good5 $$1 - 0.473T + 5T^{2}$$
7 $$1 + 4.55T + 7T^{2}$$
11 $$1 - 3.49T + 11T^{2}$$
13 $$1 + 0.0840T + 13T^{2}$$
17 $$1 - 3.61T + 17T^{2}$$
19 $$1 + 3.61T + 19T^{2}$$
23 $$1 - 2.82T + 23T^{2}$$
29 $$1 + 7.30T + 29T^{2}$$
31 $$1 - 0.557T + 31T^{2}$$
37 $$1 - 6.20T + 37T^{2}$$
41 $$1 - 9.27T + 41T^{2}$$
43 $$1 + 2.27T + 43T^{2}$$
47 $$1 + 2.82T + 47T^{2}$$
53 $$1 + 0.697T + 53T^{2}$$
59 $$1 - 5.65T + 59T^{2}$$
61 $$1 + 3.85T + 61T^{2}$$
67 $$1 - 5.33T + 67T^{2}$$
71 $$1 + 9.11T + 71T^{2}$$
73 $$1 + 0.541T + 73T^{2}$$
79 $$1 + 10.9T + 79T^{2}$$
83 $$1 + 15.0T + 83T^{2}$$
89 $$1 - 14.6T + 89T^{2}$$
97 $$1 - 4.31T + 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

−7.25462829694771830514665434111, −6.63418263904630783378391319724, −6.00957546313444182709787796636, −5.63514148948046398475165084109, −4.37862506242163473953890770591, −3.77703259601728898003175729745, −3.14157221243803111160534895048, −2.27534018618167788497511261890, −1.15913802002400196484589656062, 0, 1.15913802002400196484589656062, 2.27534018618167788497511261890, 3.14157221243803111160534895048, 3.77703259601728898003175729745, 4.37862506242163473953890770591, 5.63514148948046398475165084109, 6.00957546313444182709787796636, 6.63418263904630783378391319724, 7.25462829694771830514665434111